348 



MEMOIRS OF THE AMERICAN ACADEMY 



Infinitesimal Relatives. 

 We have by the binomial theorem by (49) and by (47), 



(1 _[_ X )n = 1 -f X p %n-P + X n . 



Now, if we suppose the number of individuals to which any one thing is x to be 

 duced to a smaller and smaller number, we reach as our limit 



x 2 



0, 



y L p x n —P 



\n\.i n 



ti x u 



? 



xn 



(i + *) 



n 



1 -f- xn . 



9 



If, on account of the vanishing of its powers, we call x an infinitesimal here and de- 



note it by i, and if we put 



xn 



in 



our equation becomes 

 (109.) 



y 



y 



(l + tX = l+y. 



/ 



Putting y = /, and denoting ( 1 + *) * b y 6 > we tave 



(110.) 



6 



r 



(1 + e)T 



1 + / 



In fact, this agrees with ordinary algebra better than it seems to do ; for / is itself 

 an infinitesimal, and 6 is 6 *. If the higher powers of / did not vanish, we should 

 get the ordinary development of 6. 



Positive powers of 6 are absurdities in our notation. For negative powers we have 



(111.) 



There are two 

 binomial theorem, 



6 

 ways of raising 6 



X 



1 



X . 



the # th power. In the first place, by the 



x)y 



1 



f>].ly~V fl + ^4 — -.ly-* 2 ,^ 2 — etc. ; 



2 



and, in the second place, by (111) and (10). 



6— *2r 



1 



xy 



It thus appears that the sum of all the terms of the binomial development of ( 1 



xy, 



after the first, is — xy. The truth of this may be shown by an example. Suppose 

 the number of /s are four, viz. T, Y", Y"', and Y"". Let us use x\ x", x'% and *"" 

 in such senses that 



xT 



xY 



ff 



X 



ff 



> 



xY 



rrr 



X 



rrr 



m 



xY 



nrr 



x' 



trt 



