322 Mb MOORE, ON A DIFFICULTY IN ANALYSIS, Sec. 



Treating the function sinx — ;); in the same manner as we have just 



1 of 



treated sin x, we get /3 = 3 and B= — . Similarly sin x — x + 



gives /3 = 5 and C= + , and thus finally 



^"^^ = ^- 1:1:3 -^ i.2.l4.5 ^^'^- 



This process is general, and may be easily applied to demonstrate the 

 theorems of Taylor and Maclaurin. 



4. The theorem of the last article is La Grange's principle, and 



was used by that analyst as the fundamental principle of his Calculus 



of Functions. By the corollary to the theorem in Article 2, it is 



clear that it is inapplicable to functions Avhose limits for the values of x 



which converge indefinitely to zero are infinitesimals of an infinitely 



high or an infinitely low order. Ax", Bxl^, Cx> being at the same limit 



infinitesimals of finite orders. There are however only two known classes 



_i 1 



of functions which have this property, viz. e ^" and -, . The limit of 



^ ^ -^ log x 



the ratio of e~~'' to x" is easily shown to be infinitely small, however 



great a may be taken. Lim,,.=o(e ^") is therefore an infinitesimal of an 



_1 

 infinitely high order, and consequently e j" cannot be represented by a 



series like Ax'' -\- B xl^ -ir Cxy -^ kc. On the contrary, the limit of the ratio 

 of = to x" may be shown to be infinitely great, however small a 



log X ■' JO' 



may be taken. Lim, = o(n ) is therefore an infinitesimal of an infi- 

 nitely low order, and therefore cannot be represented by a series such 

 as A^'' + Bx + Cxy + &CC. In either case indeed, if we assumM the 

 principle, we should, by passing to the limits of the equivalent series, 

 find an infinitesimal of an infinitely high or infinitely low order, equal 

 to a series of infinitesimals of finite orders, which would violate the 

 principle of homogeneity which exists equally in finite and infinitesimal 

 quantities. 



