Pure mathematics

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1874 Excerpt: ... ab-r a or--= b, and ab-f 6 or-T-= a. a b And so, xyz 4-a» = y, and pjrs 458= /w. We may then conclude that when the divisor is contained as a factor in the dividend, the quotient is found by omitting from the dividend those of its factors which constitute the divisor. If the divisor be not contained as an exact factor in the dividend, we may then express the quotient symbolically. Thus, xy-5-ab =---?. ab "When, however, the dividend and divisor have a common factor, it is plain that we may, as in arithmetic, strike out of the numerator and denominator of the symbolical quotient this common factor. Thus, 5 abc bd = ±£ = if?. bd d And 16 xyz-f 10 axz =--; TMz =--. 10 axz o a 28. A power of a quantity is divided by any other power of the same quantity by subtracting the index of the divisor from that of the dividend, the quotient being that power of the quantity whose index is the remainder so obtained. 1. Let the power of the quantity in the dividend be the higher. "We have a5 = aaaaa, and a3 = aaa. 5 3 aaaaa 2 s_. /. or f = = aa = «r= a ". aaa Or, generally, m being greater than n, since am--aaa...to m factors, and a" = aaa...to n factors, we have--_ _ aaa... torn factors aaa...to n factors 2. Let the power of the quantity in the dividend be the Suppose we have to divide a by a.7. aaaa 1 1 aaaaaaa aaa a "We may, however, so express the result that it shall agree exactly with the proposition at the head of this article. For we may conceive of a3 as representing the product of unity and the quantity a3. We shall therefore be perfectly consistent if we allow a3 to represent the quotient of unity by the quantity a3. We shall then have a3 =--, and hence we get from the above result--a-r a1 = a3 = a4"'. Or, genera...