Chapter XXXII: Appendix: C

A LECTURE ON THE THEORY OF SAMPLING.

The problem of the sampler is essentially the same as that of the student of statistics. One aims at getting a small parcel of ore, the other a number of data, but each hopes to obtain what shall represent a true average applicable to a much larger mass of material. Ignoring the mechanical part of the problems, the sampling errors of the one and the deviations from the average of the other are the same thing.

It may be doubted whether many not specially trained in the study of statistics could answer such a question as the following:--Seven hundred thousand men being employed, there are, in a given year, one thousand deaths from accident. Assuming the conditions to remain unaltered, within what limits could one foretell the number of deaths by accident in any other year?

On the other hand, there is a widespread belief in the efficacy of what is called the law of averages. Even the ordinary newspaper reader is accustomed to look on the national death-rate or birth-rate as a thing capable of being stated with accuracy to one or two places of decimals, and he knows that the annual number of suicides is practically constant.

If a man played whist often and kept a record of the number of trumps n each hand, he would find fortune treated him quite fairly; in a year's play the average number would deviate very little from the theoretical average, _i.e._, one-quarter of thirteen. And a knowledge of this truth is useful, and that not merely in keeping ejaculations in due restraint. But every good player knows more than this: he has a sense of what variations in the number of trumps may reasonably be expected. For example, he will be prepared to risk something on neither of his opponents having more than five trumps, and will accept it as a practical certainty that no one has more than eight. Much of what is known as good judgment is based on a proper estimate of deviations from the average. The question has an important bearing on sampling, as may be seen from the fact that shuffling and dealing at cards are but modifications of the well-known mixing and quartering of the sampler.

Because of this bearing on sampling and for other reasons, I became many years ago much interested in the question, and gave to its solution perhaps more labour than it was worth. In books on Medical Statistics the answer to the question is stated in a mathematical formula, called Poisson's formula, which, in a modified form, I shall give further on. But this did not satisfy me, because I wanted to learn what a reasonably safe _limit of error_ actually meant, and this could be best learnt by experiment; so with the help of some friends I went in for a thorough course of penny-tossing.

Tossing a penny twenty times, an average result would be ten heads and ten tails. To find the deviations from this, we tossed two hundred twenties, _i.e._, four thousand times. Of the two hundred, thirty-three gave the exact average, viz.:--10 heads; sixty-four gave an error of one, viz.:--9 or 11 heads; forty-nine, an error of two; twenty-six, an error of three; twenty, an error of four; eight gave an error of five, and this limit was not exceeded. From these we may say that six is a reasonably safe limit of error. Ninety-seven cases, say one-half, gave an error not exceeding one; and the mean error is 1.8.

In other words, in twenty tosses you will not get more than 16 nor less than 4 heads; you are as likely as not to get 9, 10, or 11 heads; and lastly, if you lost in twenty throws all heads or tails over 10 your average loss would be 1.8 penny, or say roughly 2d. on the twenty throws.

It was necessary to compare these with another series containing a larger average, say that of 100 heads in 200 throws. I confess the labour of tossing pennies two hundred at a time was little to our taste. So from a bag of pennies borrowed from the bank, we weighed out samples containing two hundred, and for an evening we were busy counting heads and tails in these. The heads in sixty samples ranged from 80 to 114. One hundred heads occurred seven times. The extent and frequency of the errors is shown in the table.

------+-------+------+-------+------+-------
Error.|No. of |Error.|No. of |Error.|No. of
| Times.| | Times.| | Times.
------+-------+------+-------+------+-------
1 | 8 | 6 | 3 | 11 | 1
2 | 5 | 7 | 3 | 14 | 3
3 | 6 | 8 | 3 | 15 | 1
4 | 3 | 9 | 7 | 18 | 2
5 | 6 | 10 | 1 | 20 | 1
--------------------------------------------

We may call the limit of error 21. Twenty-nine results out of sixty, say one-half, had an error not exceeding 4; and the mean error is 5.6. In comparing these with the series 10 in 20 we must, working by rule, divide not by 10 but by 3.16, the square root of 10; for if we multiply an average by any number[126] the error is also multiplied but only by the square root of the number. The error varies as the square root of the number. Now

21/3.16 = 6.6 = limit of error for 10 in 20.
5.6/3.16 = 1.8 = mean error " " "
4/3.16 = 1.2 = probable error " " "

It will be seen that these calculated results agree fairly well with those actually obtained. The rule by which these calculations are made is important and will bear further illustration. To calculate the number of heads in 3200 throws, we have to find the limit of error on a true average of 1600 in 3200. This being 16 times the average of 100 in 200, the corresponding errors must be multiplied by 4. This gives

21×4 = 84 = limit of error.
5.6×4 = 22.4 = mean error.
4×4 = 16 = probable error.

The results I have actually obtained with these large numbers are hardly enough to base much on, but have a value by way of confirmation. Expecting 1600 heads, the actual numbers were 1560, 1596, 1643, 1557, 1591, 1605, 1615, 1545.

It will be seen that exactly half are within the probable error; but this, considering the small number of results, must be more or less of an accident; it is more to the point they are all well within the limits of error.

I have a large number of other results which with a single exception are all in accord with those given; and this exception only just overstepped the limits. It was like a case of nine trumps, which though in a sense possible, is very unlikely to happen in any one's experience.

But even now we are not quite in a position to answer the question with which we started. If you refer to it you will see that we are face to face with this problem: the limit of variation on the 1000 who died would be say 70,[127] ignoring decimals. But if we calculate on the number who did not die, viz.--699,000,[128] we shall get a variation 26 times as great as this. But it is evident the variation must be the same in each case. I submitted this kind of problem also to the test of experiment, the results of which gave me great faith in Poisson's formula.

Imagine two hundred pennies in a bag all heads up. Any shaking will spoil this arrangement and give a certain proportion of tails. And, further, the probable effect of shaking and turning will be to reduce the preponderance of heads or tails whichever may be in excess. This of course is the reason why we are so unlikely to get more than 120 of them in either position.

But if the two hundred pennies are increased to 20,000 by adding pennies which have tails on both sides, then the shaking or mixing would be less effective. We should still expect as an average result to get the 100 heads but in 20,000 instead of 200. The variation will be 28 or 29 on the 100 instead of 20. And this is a better limit in such cases. _Taking 28 as the limit of error on 100 instances_ and proportionally increasing the others so that _the mean error becomes 7.8 and the probable error 5.6_, we may now calculate the answer without gross mistake.

The probable variation on the 1000 deaths by accident will be 18, the mean variation will be 24.6, and the limits of variation 88.5. One such table showing in five years a mean number of deaths of about 1120 per annum gives an annual deviation of about 50 up or down of this. It will be seen at once that an improvement of 30 or 40 in any one year would be without meaning, but that an improvement of from 100 to 200 would indicate some change for the better in the circumstances of the industry. Before applying these principles to the elucidation of some of the problems of sampling it will be well to give Poisson's formula (in a modified form) and to illustrate its working.

Let _x_ equal the number of cases of one sort, _y_ the cases of the other sort, and _z_ the total. In the example, _z_ will be the 700,000 engaged in the industry; _x_ will be the 1000 killed by accidents, and _y_ will be the 699,000 who did not so die. The limit of deviation or error calculated by Poisson's formula will be the square root of 8_xy_/_z_. Replacing _x_, _y_ and _z_ by the figures of the example we get the square root of (8×1000×699000)/700,000, which works out to the square root of 7988.57, or 89.3. Which means that we may reasonably expect the number of deaths not to vary from 1000 by more than 89, _i.e._, they will be between 1090 and 910. It will be seen that this number is in very satisfactory agreement with 88.5 given by the rougher calculation based on my own experiments.

To come to the question of sampling. Consider a powder of uniform fineness and fine enough to pass through an 80 sieve. For purposes of calculation this may be assumed to be made up of particles of about one-eighth of a millimetre across (say roughly 1/200 of an inch); cubed, this gives the content as about 1/500 (strictly 1/512) of a cubic m.m. Now one cubic m.m. of water weighs 1 milligram; therefore 500 such particles if they have the specific gravity of water weigh 1 milligram, and otherwise weigh 1 milligram multiplied by the sp. gr.: 500 particles of ruby silver (Pyrargyrite)[129] will weigh 5.8 milligrams and will contain nearly 3.5 milligrams of silver.

Now suppose a portion of 3.2667 grams (1/10 Assay Ton) of silver ore to contain 500 such particles of ruby silver and no other material carrying silver: such an ore would contain 35 ozs. of silver to the ton. But the limits of variation on 500 particles would be 28[130] multiplied by the square root of 5, or 62 particles. Thus the limit of sampling error would amount to just one-eighth of the silver present, or say to rather more than 4 ozs. to the ton; the mean sampling error would be rather more than a quarter of this, or say about 1.3 ozs. to the ton.

On the other hand, if one took for the assay a charge six times greater (say about 20 grams), the number of particles would be 3000 and the limits of variation would be 28 multiplied by the square root of 30, or 153 particles, which is very closely 1/20 of the silver present, or say 1.75 ozs. to the ton, whilst the mean error would amount to about .5 ozs. to the ton.

To work these examples by Poisson's formula let us assume the gangue to have a mean sp. gr. of 3. Then 500 particles would weigh 3 milligrams; and 3.2609[131] grams would contain 543,500 particles. There would be then 500 of ruby silver and 543,500 of gangue, together 544,000, and the formula gives the square root of (8×500×543500)/544000, which works out to 63 particles as against 62 by the other method.

A practical conclusion from this is of course that either the ore must be powdered more finely or a larger portion than 3 grams must be taken for the assay. Moreover, it is evident that on such an ore no small sample must be taken containing less than several million particles.

Consider now a copper ore of the same uniform fineness containing particles of copper pyrites (sp. gr. 4) of which 1000 particles will weigh 8 milligrams, mixed with gangue of which 1000 particles weigh 6 milligrams.

If one gram of such ore contain .5 gram of copper pyrites (= .1725 gram copper) and .5 gram of gangue, these will contain 62,500 and say 83,500 particles respectively. Altogether 146,000 particles. With Poisson's formula this gives the limit of sampling error as the square root of (8×62500×83500)/146000 or 521 particles. But a variation of 521 on 62,500 is a variation of .83 per cent. The percentage of copper in the ore is 17.25 per cent., and .83 per cent. of this is .14 per cent. The limits of sampling error, therefore, are 17.11 per cent. and 17.39 per cent. Again, it must be remembered that the mean sampling error would be a little over one-quarter of this, or say from 17.2 per cent. to 17.3 per cent. The practical conclusion is that a powder of this degree of fineness is not fine enough. In the last place let us consider a similar iron ore containing 90 per cent. of hæmatite (sp. gr. 5) and 10 per cent. of gangue (sp. gr. 3), 1 gram of such ore will contain 90,000 particles of hæmatite weighing .9 gram and containing .63 gram of iron with say 16,500 particles of gangue weighing .1 gram. Altogether 106,500 particles.

Poisson's formula then gives the limits of variation as the square root of (8×90000×16500)/106500 or 334 particles. But 334 on 90,000 is 0.23 on 63.0, which is the percentage of iron present. The limits of sampling error then are 62.77 per cent. and 63.23 per cent. and the mean variation is from 62.94 per cent. to 63.06 per cent.

These examples are worthy of careful consideration, and it must be remembered that the calculations are made on the assumption that the ore is made up of uniform particles of mineral of such fineness as would pass easily through an 80 sieve, but which does not pretend to represent with great exactness the fineness of the powdered ore customary in practice. They show that having passed through such a sieve is no proof of sufficient powdering, not that all ores powdered and so sifted are unfit for assaying. This last would be an absurd and illogical conclusion.

If an ore be powdered to a fairly fine sand and then be passed through a series of sieves, say a 40, 60, and 80, in such a state that little or none remains on the first, but the others retain a large proportion; then of that which comes through the 80 sieve, perhaps two-thirds by weight may be even coarser than the powder I have used in the example. Of the rest most may be of about half this diameter; the weight of the really fine powder may be quite inconsiderable. On the other hand, if the grinding be continued until, on sifting, little or nothing that is powderable remains on the sieves; then in the sifted product the proportions will be very different. This last, of course, is the only right way of powdering. Also it is evident that so much depends on the manner of powdering that nothing precise can be stated as to the average coarseness of the powder. Suppose, however, by good powdering a product is obtained which may be represented by a uniform powder with particles 1/20th of a millimetre in diameter (say roughly 1/500 inch). Compared with the previous powder, the diameter has been divided by 2.5; their number, therefore, in any given weight has been increased by the cube of 2.5, which is 15.6. But the value of a sample varies as the square root of the number of particles. Hence the reduction in size and consequent increase in number has made the sample nearly four times better than before; and it will be seen that this brings the sampling error within tolerable limits.

There are one or two words of warning which should be given. In the first place, using a 90 sieve instead of an 80 must not be too much relied on; the powder I took in the example would pass through it. It is a question of good powdering rather than of fine sifting. In the second place, a set of, say half-a-dozen, assays concordant within 1 oz. where the theory gives 4 ozs. as the limit of error does not upset the theory: the theory itself states this as likely. It is the error you _may_ get in one or two assays out of a hundred, not the error you are _likely_ to get in any one assay, which is considered under the heading "limit of error."

Accepting the result just arrived at that a portion of 1 gram may be safely taken for an assay if the particles are 1-20th of a millimetre in diameter, the further question remains as to what weight of the original sample must be reduced to this degree of fineness. This may be answered on the principle that the same degree of excellence should be aimed at in each of a series of samplings. This principle is illustrated in the table on page 2.

A fine sand, such as would pass a 40 sieve but be retained on a 60 sieve, would be fairly represented by particles one-quarter of a millimetre in diameter. This being five times coarser, to contain the same number of particles must be 125 times (the cube of 5) as heavy; therefore 125 grams of it can be taken with the same degree of safety as 1 gram of the finer powder. Of such a sand about this weight should be taken and reduced to the finer powder. If the ore were in coarse sand, say in particles 1 millimetre in diameter, this would be four times as coarse as that last considered, and we should have to take 64 times as much of it: 64 times 125 grams is 8 kilos, or say roughly from 15 to 20 lbs. This should be crushed to the finer size and mixed; then from 100 to 150 grams should be taken and ground to the finest powder.

There is, however, a reason why, on the coarser stuff, a smaller proportion may safely be used. This becomes more evident if we consider a still coarser sample. A heap of ore in stones about 2 inches across would be 50 times coarser than the sand, and an equivalent sample would need to be 125,000 times heavier; this would amount to about 1000 tons. Experienced samplers would say that under such conditions so large a sample was hardly necessary.

This is because I have assumed in the calculations that the grains of copper pyrites, for example, were all copper pyrites and the particles of gangue were free from copper. This would be true or nearly so for the very fine powder, but far from true in the case of the ore heap. In the heap probably few of the stones would be pure ore and still fewer would be free from copper. The stones would differ among themselves in their copper contents only within certain comparatively narrow limits. And it is evident that, if replacing one stone by another, instead of resulting in the gain or loss of all the copper one or other contained, merely affected the result to one-tenth of this amount, then a sample of 1-100th of the weight (say 10 tons) would be equally safe.

It should be remembered, however, that while the man who samples on a large scale can safely and properly reduce the size of his samples on this account, yet the principle is one which counts less and less as the stuff becomes more finely divided, and ought to be ignored in the working down of the smaller samples which come to the assayer.

FOOTNOTES:

[126] The 10 in 20 multiplied by 10 = 100 in 200.

[127] Multiply the errors for 100 by the square root of 10.

[128] Multiply the errors for 100 by the square root of 6990.

[129] Sp. Gr. 5.8. Silver 60 per cent.

[130] Taking 28 as the limit of variation on 100.

[131] The weight of the ore less the weight of ruby silver in it.

INDEX.

Acid measures, 49

Acidimetry, 323

Acidity of ores, 168

Acids, 54
strength of, 54, 75, 436

Air of mines, carbonic acid in, 428

Alkalies, 330
determination of, 331
Lawrence Smith's method for, 333, 412
separation of, 332

Alkalimetry, 323

Alkaline earths, 320

Alumina, 314
determination of, 315
in mineral phosphates, 316
separation of, 314, 316

Amalgamation, 126

Ammonia, detection of, 341
determination of, 342
in natural waters, 353

Antimony, 225
detection of, 227
dry assay for, 226
gravimetric assay, 228
separation of, 228
volumetric assay, 229

Arsenic, 381
detection of, 381
dry assay for, 382
gravimetric assay, 383
in brimstone, 393
in crude arsenic, 388, 393
in mispickel, 125, 392
iodine, assay for, 386
separation by distilling, 384
uranium acetate, assay for, 389
Volhard's method applied to, 124

Assay book, 11
note, 12
results, 7
tons, 13, 131

Assaying, 1
methods, 15

Assays, check, 154
preliminary, 147

Atomic weights, 69
table of, 433

Barium, 326

Baryta, 326

Barytes, sulphur in, 378

Base bullion, sampling of, 157

Basic acetate separation, 233

Baumé's hydrometer, 77

Beryllia, 319

Bismuth, 220
colorimetric assay, 223
detection of, 221
gravimetric determination of, 222
in commercial copper, 208
separation of, 222

Black tin, 271
an analysis of, 287
assay of, 276
copper in, 204
examination of, 285
separation by vanning, 272

Blank assays, 34

Blende, sulphur in, 375
zinc in, 266

Book, assay, 11
laboratory, 10
sample, 9

Boracic acid. _See Boron_

Borax, examination of, 431

Boron, 429
direct determination of, 431

Brass, copper in, 194
zinc in, 265

Bromine and bromides, 361

Bronze, copper in, 194
tin in, 281

Burettes, 51

Burnt ore, silver in, 116, 118
sulphur in, 377

Cadmium, 269
gravimetric determination, 269
separation of, 269

Caesium, 339

Calcination, 22, 92, 139, 345

Calcium, 320
detection of, 321
gravimetric determination, 321
separation of, 321
titration with normal acid, 322
titration with permanganate, 322

Calculation of results, 7

Calculations from formulæ, 70

Calorific effect of coal, 419

Calorimeter, 419

Calx, 345

Carbon, 414
gravimetric determination, 416
in iron or steel, 423

Carbonates, 424

Carbonic acid in the air of mines, 428

Caustic potash = potassium hydroxide, 65

Caustic soda = sodium hydroxide, 66

Cerium, 318

Chalybite, iron in, 243

Charcoal, 21, 94

Check assays for gold, 154
for silver, 104, 113

Chlorine and chlorides, 359

Chromium, 307
gravimetric assay, 309
in chrome iron ore, 308
volumetric assay, 309

Clays, examination of, 316

Coals, 418

Cobalt, 259
detection of, 259
dry assay for, 251
gravimetric determination, 260
in hardhead, 288
separation from nickel, 442, 254, 258

Coke, 25

Common salt, examination of, 336

Concentrates, assay for gold of, 140

Colorimetric assays, 44

Copper, 175

Copper, bismuth in, 208
colorimetric assay for, 190, 203
commercial, arsenic in, 208, 388
commercial, copper in, 193
commercial, examination of, 205
cyanide assay for, 194
dry assay of, 176
dry assay, loss of, in, 176
electrolytic assay for, 190, 203
gold in, 206
iodide assay for, 199
iron in, 209, 249
lead in, 206
separation of, 183
silver in, 205
sulphur in, 207

Copper ores, solution of, 183
valuation of, 181

Copper pyrites, copper in, 179, 188, 198, 202
sulphur in, 376

Culm, 22

Cupel, 23, 142

Cupellation, loss, corrections for, 103
loss in gold, 145
loss in silver, 101
of gold lead alloys, 182
of silver lead alloys, 98, 110
temperature of, 143

Cyanicides, 169

Cyanide assay for copper, 194
for nickel, 255
for tin, 278

Cyanides, alkalinity of, 167
assay of, 167
commercial, 160
double, 161
gold-dissolving power, 162
prussic acid, 162
volumetric determination of, 163, 165

Cyanide liquors, alkalinity of, 167
assay of, 164, 165
assay of, for gold, 140
assay of, for zinc, &c., 169

Daniell cells, 185

Didymium, 319

Dollars to the ton, 9

Dry assays, 16

Drying, 5, 33

Earths, 314
the alkaline, 320

Electrodes, 187

Electrolysis for copper, 184
for nickel, 254

Equations, 69

Erbia, 319

Ferrous and ferric salts, 231

Filtration, 31

Finishing point, 42

Flasks, graduated, 49

Flatting, 149

Fluorine and fluorides, 363

Fluxes, 16, 93, 136, 138, 140

Formulæ, 68

Furnaces, 25

Galena, lead in, 217, 218

Gangue, 405
iron in the, 244

Gas-measuring apparatus, 52

Gases, measurement of, 44

Gay-Lussac's assay for silver, 119
assay for silver modified, 123

German silver, copper in, 194
nickel in, 255, 259

Gold, 126
amalgamation of, 126
in cyanide liquor, 140
loss of, in cupellation, 145
loss of, in parting, 154
preparation of, 63
silver in, 157
silver in, after parting, 154
test for, 126

Gold-lead alloys, cupellation of, 142
sampling of, 158

Gold ores assay with cyanide solutions, 141
calcination of, 139
concentrates, 140
fluxing, 136, 138, 140
sampling of, 127
size of assay charges, 127
tailings, 140

Gold-parting, 150
platinum in, 145, 154, 170, 171

Gold-zinc slimes, 142

Graduated vessels, 49

Gravimetric methods, 15, 27

Halogens, 358

Hardhead, 287
an analysis of, 289

Hot plate, 30

Hydrogen, preparation of, 62
reduction by, 280

Hydrometer, 77

Ignition, 32
in hydrogen, 280

Indicators, 42

Inquartation, 146

Iodine and iodides, 362

Iridium, 171

Iron, 231
bichromate assay for, 237, 243,
carbon in, 423
colorimetric assay for, 247
ferrous and ferric, 231
gravimetric determination, 233
permanganate assay for, 236, 238
phosphorus in, 399
reduction of ferric solutions, 235, 241
separation of, 232
stannous chloride assay for, 244
volumetric assays for, 234

Iron ores, iron in, 244, 247
phosphates in, 399

Laboratory books, 9

Lanthanum, 319

Lawrence Smith's method for alkalies, 333, 412

Lead, 211
colorimetric assay for, 218
detection of, 211
dry assay for, 211
gravimetric determination of, 213
in commercial copper, 206
in commercial zinc, 214
in galena, 217, 218
separation of, 211, 213
volumetric determination of, 214

Litharge, use of, in dry assays, 20, 93

Lithium, 338

Lime, 320
milk of, 321
volumetric assays for, 322

Limestone, examination of, 329
lime in, 324

Limewater, 321

Loths, 9

Magnesia, magnesium, 328
mixture, preparation of, 64

Manganese, 298
colorimetric assay, 306
detection of, 299
gravimetric determination of, 300
separation of, 299
volumetric determination of, 300

Manganese peroxide, ferrous sulphate assay for, 301
iodine assay for, 302
= manganese dioxide, 298

Manganese ore, copper in, 204
manganese in, 300
peroxide in, 302

Matte, 18

Measuring, 49
flasks, 49
gases, 44, 52
gold buttons, 133, 440,
liquids, 49
silver buttons, 106

Mechanical methods, 16

Mercury, 171
dry assay, 172
wet assay, 173

Metallic particles in ores, gold, 129
particles in ores, silver, 108
particles, tin, 278, 287

Micrometer, 133

Microscope, measuring with the, 440, 133

Mispickel, arsenic in, 125, 392
sulphur in, 376

Moisture, 7, 350

Molybdate separation for phosphates, 395
solution, preparation of, 60

Molybdenum, 311

Muffle, 25

Nessler's solution, 342

Nickel, 251
dry assay for, 251
electrolytic assay, 254
gravimetric determination of, 254
in German silver, 255, 259
separation from cobalt, 254, 258, 442
separation from iron, 258
separation from manganese, 258
separation of, 253
volumetric assay, 255

Niobium, 297

Nitre, 22
use of, in dry assays, 95

Nitrogen and nitrates, 400

Nitrometer, 403

Normal acid, normal solutions, 323

Ores, determining water in, 5, 351
drying, 5
powdering, 4, 109, 130, 448
quantities of, for an assay, 11, 27, 127
sampling, 1, 127, 444
with metallic particles, 3, 108, 129

Osmiridium, 171

Osmium, 171

Ounces to the ton, long, 107
to the ton, short, 132

Oxidation, 345

Oxides, 345
determination of oxygen in, 346

Oxidising agents, 22, 95, 345
effect of nitre, 95
effect of nitric acid, 56

Oxygen, 344
equivalent, 358
in natural waters, 344, 356
in ores, 348

Palladium, 171

Parting, 150
acids, 150
in flasks, 151
in glazed crucibles, 153
in special apparatus, 156
in test tubes, 152

Phosphate, assay of apatite for, 399
assay of iron ore for, 399

Phosphates, gravimetric assay, 396
volumetric assay, 397

Phosphorus and phosphates, 394
in iron, 399

Pipette, 50, 120

Platinum, 170
in gold, 145, 154, 170

Potash, commercial examination of, 338

Potassium, 336
gravimetric determination, 337

Potassium cyanide, 22, 65, 160
commercial assay of, 167
commercial, purity of, 161

Powdering, 4, 130, 448, 109

Precipitation, 30

Precipitates, drying, 32
igniting, 32, 34
washing, 31

Preliminary assays, 104, 147

Preparation of acids, 54
of other reagents, 59

Prill, 108, 129, 278, 287

Produce, 8

Pyrarsenate of magnesia, 383

Pyrites, iron in, 244
sulphur in, 370, 376

Pyrophosphate of magnesia, 397

Quantity to be taken for an assay, 11, 27, 127

Quartation, 146

Quartering, 2

Reagents, strength of, 54

Red lead for dry assays, 20, 22, 94

Reducing agents, 21, 94
effects of charcoal, &c., 94
effect of mineral sulphides, 95, 97, 98

Reduction by hydrogen, 280
of ferric solutions, 235, 242, 244

Regulus, 18

Report form, 12

Results, calculation of, 7, 13, 16, 38, 107, 131, 132
statement of, 7

Rhodium, 171

Roasting, 22, 345

Rolling, 149

Rubidium, 340

Ruthenium, 171

Sample book, 9

Sampling, 1
effect of powdering on, 449
errors, 447
gold ores, 127
metals, 157
theory of, 444

Scorification of silver ores, 88

Scorifier, 23, 89

Selenium, 379

Separation, as sulphides, 57
basic acetate, 233
molybdate, 395

Shales, bituminous, 420

Silicon and silicates, 405
in iron, 414

Silica in rocks, 409
in slags, 414

Silicates, alkalies in, 333, 412
beryllia in, 320
examination of, 409
titanium in, 411

Silver, 87
correction for cupellation loss, 103
detection of, 87
Gay-Lussac's assay, 119
Gay-Lussac's assay modified, 123
gravimetric determination of, 117
in bullion, 113
in burnt ore, 116, 118
in copper, 114, 205
in galena, 114
in lead, 113
in oxide of lead, 113
in silver precipitate, 115
loss in cupellation, 101
pure preparation of, 66
Volhard's assay, 121
volumetric methods, 119, 121, 123

Silver lead alloys, cupellation of, 98
sampling of, 157

Silver ore, crucible assay of, 90
metallic particles in, 108
scorification of, 88

Size of assay charges, 11, 27, 127

Slags, 19

Soda-lime, 425

Sodium, 334

Sodium cyanide, 160

Solution, 29

Solutions, normal, 323
standard, 36

Specific gravity, 75, 436

Speise, 19

Standard, 37
solutions, 36

Standardising, 37

Steel, carbon in, 423
chromium in, 310
manganese in, 300

Stoking, 25, 143

Strength of reagents, 54

Strontium, 324

Sulphates and sulphur, 367
gravimetric determination, 369
volumetric determination, 370

Sulphides, reducing action of, 9, 95

Sulphocyanate assay for silver, 121

Sulphur in blende, 375
in burnt ore, 377
in chalcocite, 376
in coal, 419
in copper, 207
in copper pyrites, 376
in mispickel, 376
in pyrites, 370, 376

Sulphuretted hydrogen, preparation, 57

Surcharge, 154

System in assaying, 28

Table, atomic weights, 433
comparing thermometers, 435
ounces to the long ton, 107
ounces to the short ton, 132
sp. g. ammonia, 438
sp. g. hydrochloric acid, 437
sp. g. minerals, 86
sp. g. nitric acid, 436
sp. g. sulphuric acid, 439
sp. g. water, 83

Tantalum, 297

Tartar, 20, 94

Tellurium, 379
improved test for, 150

Thallium, 219

Thorium, 317

Tin, 271 _See also Black tin_
assay for, by vanning, 273
copper in, 204
Cornish assay, 276
cyanide assay, 278
detection of, 279
gravimetric determination of, 281
iron in, 250
separation of, 280
volumetric assay for, 282

Tin arsenide, 284

Tin phosphide, 284

Tin slag, 290
an analysis of, 292
tin in, 290

Titanium, 292
detection of, 293
in black tin, 272, 287
in rocks, 411
separation, &c., 294

Titration, 35
indirect, 43, 72

Ton, assay, 13, 131,
long, 2240 lbs. = 32,666.6 oz., 107
short, 2000 lbs = 29,166.6 oz., 132

Tungsten, 295

Tungstic acid, 295
gravimetric determination, 296
in black tin, 285
in wolfram, 296

Uranium, 312

Valuation, of copper ores, 181

Vanadium, 310

Vanning, 273

Volhard's assay applied to arsenic, 124
silver assay, 121

Volume-corrector, 53

Volumetric assay, 35, 38

Water, 7, 350
direct determination of, 351
examination of, 352
expansion of, 83
solids in, 354

Weighing, 47
small gold buttons, 131

Weights, 47

Wolfram, an analysis of, 296
tungstic acid in, 296

Yttria, 319

Zinc, 261
commercial, examination of, 268
commercial, iron in, 249
commercial, lead in, 214
dry assay, 261
gasometric assay, 266
gravimetric determination, 262
in blende, 266
in cyanide liquors, 169
in silver precipitate, 266
separation of, 262
volumetric assay, 263

Zirconia, 317

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THIRD EDITION, _Revised, with an Additional Chapter on Foundations. Numerous Diagrams, Examples, and Tables. Large 8vo. Cloth. 16s._

~THE DESIGN OF STRUCTURES:~

~A Practical Treatise on the Building of Bridges, Roofs, &c.~

BY S. ANGLIN, C.E.,

Master of Engineering, Royal University of Ireland, late Whitworth Scholar, &c.

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~A PRACTICAL TREATISE ON~

~BRIDGE-CONSTRUCTION:~

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GENERAL CONTENTS.--PART I.--Elementary Statics. PART II.--General Principles of Bridge-Construction. PART III.--The Strength of Materials.