11 4 Prof. J. R. Young on the Remainder of the Series 



(1 +;r)-3= 1 -3^ + 6.r2-10x3+ ...A3(-j:)'"-» + A3R1 

 : +A2R2+R3 



{l+a!)-=l-nx+'^^^K^-8ic...A„{-xy'^-' + A^'R, 



1 • ^ 



+ A„_iR2 + A„_2R3+... + AiR„, 

 the A, being introduced before R„ for the sake of uniformity 

 of notation: its value is evidently unit. When R,. is replaced 

 by {l+x)~''{—xy"'f this result becomes the same as that in 

 Professor Graves's paper ; and it must certainly strike a reader 

 as a circumstance worthy of notice, that an expression thus 

 obtained by aid of only the first principles of algebra, should 

 virtually involve a theorem of such interest in the higher re- 

 searches of analysis as that given in the paper alluded to. 



I fear, from the remark in the first paragraph of that paper, 

 that I must have expressed myself somewhat obscurely in re- 

 ference to the Calculus of Operations making " no provision 

 for the correction " I had adverted to as necessary. I think 

 I ought to have added, that this provision should always be 

 furnished by the theorem for quantity whence that for opera- 

 tions is derived. 



I take this opportunity of mentioning that the general form 

 of the theorem respecting squares, namely, 



which the Rev. Mr. Kirkman has done me the favour to insert 

 at page 500 of the last volume of this Journal, I should prefer 

 to have appeared in the following more comprehensive shape : 



to which may be added the analogous theorems 



where m and n are any positive whole numbers whatever. 

 Belfast, Jan. 13, 1849. 



Note. I submitted the substance of the foregoing investiga- 

 tion to Professor Graves, who, in reply, did me the favour to 

 communicate to me a sketch of two other methods of arriving, 

 algebraically, at the same result : this I here give in his own 

 words : — 



* * * « I indicated the method of obtaining the re- 

 mainder by differentiation, because that process admits of 

 being described in the fewest words, though it is far from 

 being the simplest. I know of two algebraical methods by 

 which the result is obtained more easily. 



