for all values of x, and for any particular value of n. Change x into x + 1; then +(-1)+1 x+r n+1 = x+n+1 x (x+1). . (x + n + 1) ' But nCr+nCr-1=n+ıCr, for all values of r [Art. 242]. Hence if the theorem be true for any particular value of n it will be true for the next greater value. But the theorem is obviously true for all values of x when n=1: it is therefore true for all positive integral values of n. By giving particular values to x we obtain relations between Co, C1, &c. For example: coac1 (a− 1)+c2 (a − 2) – cz (a − 3) + ...... + ( − 1)” c2 (a — n)=0, n (n − 1) (n − 2) (n - 1) (n-2) 1.2 Multiply (i) by a and (ii) by n and add; then a-n (a− 1) + ...(ii). a-1-(n-1) (a − 2) + +(-1)n-1 (an)=0............(iv). Now multiply (iii) by a and (iv) by n and add; then 256. Continued product of n binomial factors of the form x + a, x + b, x+c, &c. the sum of all the It will be convenient to use the following notation: S is written for a+b+c+ letters taken one at a time. S is written for ab + ac+ the sum of all the products which can be obtained by taking the letters two at a time. And, in general, S, is written for the sum of all the products which can be obtained by taking the letters r at a time. Now, if we take a letter from each of the binomial factors of (x + α) (x + b) (x + c) (x + d).............., and multiply them all together, we shall obtain a term of the continued product; and, if we do this in every possible way, we shall obtain all the terms of the continued product. We can take a every time, and this can be done in only one way; hence " is a term of the continued product. We can take any one of the letters a, b, c..., and x from all the remaining n-1 binomial factors; we thus have the terms ax"-1, bx"1, cx-1, &c., and on the whole n-1 Again, we can take any two of the letters a, b, c......, and from all the remaining n-2 binomial factors; we thus have the terms aba"-2, acx^-2, &c., and on the whole S2.x-2. n-2 And, in general, we can take any r of the letters a, b, c..., and x from all the remaining n-r binomial factors; and we thus have S. ∞". Hence (x+a)(x+b) (x+c)................ n-1 = x2+S1.x2 ̄1+S2, x22 +...+S,.x"++..., the last term being abcd......, the product of all the letters a, b, c, d, &c. By changing the signs of a, b, c, &c., the signs of S1, S, S, &c. will be changed, but the signs of S, S4, S &c. will be unaltered. n-1 -2 =x" - S1.x"1+S,.x" — ...+(-1)'S,a"...+(-1)"abcd.... 2° 257. Vandermonde's Theorem. The following proof of Vandermonde's Theorem is due to Professor Cayley*. [See also Art. 245.] We have to prove that if n be any positive integer, and a and b have any values whatever; then will Assume the theorem to be true for any particular value of n. Multiply the left side by a +b-n; it will then become (a+b)+ Multiply the successive terms of the series on the right also by a+b-n but arranged as follows: for the first term for the second {(a - n) + b}; {(a − n + 1) + (b − 1)} ; and for the rth {(a− n + r− 1) + (b − r + 1)}. We shall then have (a+b) n+1= an{(a− n) + b} + „C1 • α-1 b1 {(a − n + 1) + (b − 1)} Now n + C2. an-2 b2 {(a− n + 2) + (b − 2)} + ... +nCr_1 • an-r+1 br-1 {(a − n + r− 1) + (b − r +1)} - an {(a — n) + b} = an+i+an b12 nC1 · ɑn-1 b1 {(a − n + 1) + (b − 1)} = nС1 (ɑn b1 + ɑn_1 ba), nCr-1 • αn-r+1 br_1 {(a − n + r − 1) + (b − r + 1)} n = = nCr-1 (αn_r+2 br_1 + ɑn-r+1 br), C. a, b, {(an + r) + (b − r)} = n Cr (an-r+1 br+an-r br1a) b2 {a + (b − n)} = α2 bn + b2+1° * Messenger of Mathematics, Vol. v. Hence (a+b) n+1 = αn+1 + (1 + nC1) αn b1 + since +(nCr-1+nCr) an+1_r by + ... + b2+1 = + C1 am b1 + ... + + 1 Cr anti-r by + ... + b2+12 n+1 n 1 'n+1-r "Cr-1+nCr=n+1 Cr. Thus, if the theorem be true for any particular value of n, it will also be true for the next greater value. But it is obviously true when n=1; it must therefore be true when n = 2; and so on indefinitely. Thus the theorem is true for all positive integral values of n. 258. The Multinomial Theorem. The expansion of the nth power of the multinomial expression a+b+c+... can be found by means of the Binomial Theorem. For the general term in the expansion of (a+b+c+d+...)", that is of {a + (b + c + d + ...)}", by the Binomial Theorem is The general term in the expansion of (c+d+...)"-r-' by the Binomial Theorem is |