[ 113 ] '■'•■■''* 



XIII. Ofi the Remainder of the Series in the development of 

 (l+ar)"", and on a Theorem respecting the products of 

 Squares. By J. R. Young, Professor of Mathematics^ Bel- 

 fast^. 



IN the last Number of the Philosophical Magazine, there is 

 a very interesting paper, by Professor Graves, On the 

 Calculus of Operations, in which he has communicated a valu- 

 able theorem in that important department of analysis, which 

 I believe has not hitherto appeared in a complete form. 



Professor Graves has been enabled to deduce this theorem 

 from the previous development of (1 +^)~"; which, by means 

 of the differential calculus, he has exhibited in connexion with 

 the remainder of the series. 



This completed form of the expansion may be readily ob- 

 tained by a process imitative of that employed in my paper 

 published in the November Number of this Journal, and with- 

 out involving any operation of a more advanced character than 

 that of common algebraical division. It is as follows: — 



As ip Professor Graves's notation, let 



. _ w(n+l)....(n + ?^ — 2) 

 " 1.2 (m-1) 



_ m(?M+ 1) . . . .m + n— 2 



~ 1.2 (w-1) ' 



Put also 



(H-j;)-'(-a;)'» = R„ (H-^)-2(-a^)- = R2, 



(l+:r)-3(-^)'« = R3,&c.; 

 then, since 



( 1 +^)-» = 1 -.r + a^2_gjc ( _.r)'»-i + R„ 



we shall have, by dividing the terms on the right severally by 

 (1 •\-x\ the following rows of results, namely, 



(1 +:r)-2= 1 —xJf.x^-a^-\- (-a?)'«-' + Ri 



— :r + x2— a?a + (_^)«»-» + Rj 



■^x^—a?-\- {-xY-^ + Ri 



-0^ + (-a;)'»-»+R, 



&c. &c. +R2; 



that is, 



(1 +.r)-2= 1 -2a?4- 3a?2_4.a^ + . . . . A2(-a?)'"-» + A2R1 + Rg. 



Similarly, 



* Communicated by the Author. 

 Vhil. Mag. S. 3. Vol. 34-. No. 227. Feb. 1 84-9. I 



