A = AB, (1) B = BC; (2) hence A = ABC, (3) Thus the syllogism Darii does not really differ from Barbara. If the reader prefer it, we can readily employ a distinct symbol for the indefinite sign of quantity. Let P = some, Q = metal, B and C having the same meanings as before. Then the premises become PQ = PQB, (1) B = BC; (2) hence, by substitution, as before, PQ = PQBC. (3) Except that the formulæ look a little more complicated there is no difference whatever. The mood Ferio is of exactly the same character as Darii or Barbara, except that it involves the use of a negative term. Take the example, Bodies which are equally elastic in all directions do not doubly refract light; Some crystals are bodies equally elastic in all directions; therefore, some crystals do not doubly refract light. Assigning the letters as follows:-- A = some crystals, B = bodies equally elastic in all directions, C = doubly refracting light, *c* = not doubly refracting light. Our argument is of the same form as before, and may be concisely stated in one line, A = AB = AB*c*. If it is preferred to put PQ for the indefinite *some crystals*, we have PQ = PQB = PQB*c*. The only difference is that the negative term c takes the place of C in the mood Darii. *Ellipsis of Terms in Partial Identities.* The reader will probably have noticed that the conclusion which we obtain from premises is often more full than that drawn by the old Aristotelian processes. Thus from “Sodium is a metal,” and “Metals conduct electricity,” we inferred (p. 55) that “Sodium = sodium, metal, conducting electricity,” whereas the old logic simply concludes that “Sodium conducts electricity.” Symbolically, from A = AB, and B = BC, we get A = ABC, whereas the old logic gets at the most A = AC. It is therefore well to show that without employing any other principles of inference than those already described, we may infer A = AC from A = ABC, though we cannot infer the latter more full and accurate result from the former. We may show this most simply as follows:-- By the first Law of Thought it is evident that AA = AA; and if we have given the proposition A = ABC, we may substitute for both the A’s in the second side of the above, obtaining AA = ABC . ABC. But from the property of logical symbols expressed in the Law of Simplicity (p. 33) some of the repeated letters may be made to coalesce, and we have A = ABC . C. Substituting again for ABC its equivalent A, we obtain A = AC, the desired result.