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.[** PP: page reference]