equivalents of the respective substances from which the per-centage strength may be found by the rule of proportion, as before explained. Thus, in the case of a 100-gr. sample of carbonate of soda which has lost 15-1/4 gr. of carbonic acid, by the assay, this would be—— 22 : 31 :: 15-1/4 : 21·48% or nearly 21-1/2 per cent. of pure soda. If 53, the equiv. of anhydrous carbonate of soda, be taken, instead of 31 (the eq. of pure soda), the answer would have been, in the terms of that substance, 36·748%, or nearly 36-3/4 per cent. When an aliquot part of 100 gr. has been taken for the assay, either the result, or the third term of the proportion, must, of course, he multiplied by the denominator and divided by the numerator of the fraction representing such aliquot part. By multiplying the weight of carbonic anhydride lost, by the numbers opposite the names of the respective alkalies and their carbonates in the second column of the following _Table_ the equivalent per-centage value of the carbonates examined may be obtained in terms corresponding to the various denominations named therein, when 100 gr., or any aliquot part of 100 gr., have been tested; the result, in the latter case, being, of course, multiplied as before. By taking certain standard weights for the assay, the quantity of carbonic acid evolved may be made to furnish the per-centage strength or value of the specimen in the terms of either the pure or carbonated alkalies, whether in their anhydrous or hydrated state. The numbers in the second column of the following _Table_ represent the quantity in grains and decimal parts of each of the substances named in the first column, equivalent to one grain of carbonic anhydride. These numbers, as already mentioned, may be employed as factors for converting any numbers representing grains of that acid into the equivalents of these substances, true to 4 places of decimals; and further, they furnish us with the data for determining the exact number of grains which must be tested, so that the loss of weight in carbonic anhydride shall at once give us the per-centage richness of the sample in the terms of the denomination for which it is taken. The numbers in the third column of the _Table_, formed by simply moving the decimal point of the numbers in the second column one figure further to the right, indicate the weights to be taken for the assay, so that the loss of weight, reckoned in tenths of a grain, exactly represents the per-centage strength in the terms sought. The weights corresponding to the numbers in the fifth column give the same results, provided the loss of weight is reckoned in quarter-grains; those in the sixth column effect the same when the loss of weight is reckoned in half-grains; whilst those in the last column require that the gas eliminated should be counted in grains, and are simply the numbers in the second column of the _Table_ multiplied by 100, or reproduced by moving the decimal point two figures to the right.