In _commercial assays_, when 100 gr. (or some aliquot part thereof) are taken for trial, the per-centage result is obtained from the number of alkalimeter-divisions, or the number of grains, of the test-acid consumed, by the common Rule of Proportion. Thus:——A crude sample of potash having taken 90 alkalimeter-divisions of test-acid to neutralise it, would contain—— 100 : 47 :: 90 : 42·30% or nearly 42-1/3 per cent. of pure potassa. If only 50, 25, or 20 gr. are tested, the result must, of course, be double, quadruple, &c., as the case may be. Or the third term of the proportion may be multiplied by the denominator of the fraction representing the aliquot part. This, in the case of 50 gr. (repeating the above example), would be—— 10 : 47 :: 45 × 2 : 42·30% as before; but even these easy calculations may be simplified, as is shown below. One of the advantages, and not the least, attending the use of test-acids corresponding to equivalents, is, that by means of the simple Rule of Three, the per-centage quantity of alkali may be found whether 100 or any other number of grains have been submitted to trial. For——The weight of the sample tested (in grains) bears the same relation to the equivalent weight of the alkali under examination, that the number of alkalimeter-divisions or of the grains of test-acid consumed do to the per-centage of alkali sought. Thus, with a sample of 33 gr. of pearlash taking 35 alkalimeter-divisions or 350 grains (every 10 gr. being = 1%) of test-acid for neutralisation, this would be—— 33 : 47 :: 35 : 49·85% or nearly 50 per cent. of pure potassa. By substituting the equiv. of the dry carbonate of potash (69), for that of pure potassa used above, the quantity of that article corresponding to the same weight of the pure alkali may be at once found. Repeating the last example this will be—— 33 : 69 :: 35 : 73·18% or nearly 73-1/4 per cent. The same applies to all the alkaline bases and their carbonates. For commercial purposes, there is used, amongst others, an empirical solution, as a test-acid for potassa, soda, and ammonia, to save the necessity of calculation. This is dilute sulphuric acid having a sp. gr. of about 1·071; 100 alkalimeter-divisions (1000 water-grains measure) exactly saturate 100 gr. of pure potassa, or 113 gr. of anhydrous carbonate of soda. The number of measures consumed, read off by mere inspection from the scale of the alkalimeter, gives the exact per-centage of alkali in the sample examined, for POTASH; and by multiplying it by ·66, that for SODA also. By employing ·362 as the multiplier, it gives the like result for AMMONIA. In fact, occasionally, in order to save the necessity of any calculation, two ‘test-acids’ are frequently employed——the one for potash and the other for soda.