of certain Branches of Analysis. 347 



the evanescent arc at right angles to the diameter ; and hence 

 also must it coincide with the evanescent x or <^ x. Conse- 

 quently fl = 1. The factorial expressions may, therefore, be 

 developed into the following series 



<px = X — AgX^ + AjX* — A^x'' + &c. 

 4/^=1- A^x- + A,x* - Agx^ + &c. 

 where Aj A3 A^ &c. are known functions of tt and numbers. 



The analysis of 4> and 4/ might now be deemed complete ; 

 but there are evident means of reducing the series to still sim- 

 pler forms. The circumstances which regulate the invaria- 

 bility of triangles give us the following formula of connexion 

 between ^j and 4'j which, it is likely, will lead to important 

 consequences. 



^ J±--f^ 7r+ z} = ±{-\T^^. (1) 



The development of f (x + z) is 



<f X + ^'x. 1 + 9"x ^2 + <f"'x ^^ + &c. (2) 



And it is plain that when this series becomes equivalent to 

 ± "^'i, the coefficient of 2; or <|>'x must be equal to zero. This 



is the case when ^^ __ _^ '2.n+ \ ^ 



— 2 

 which expression must therefore include at least one class of 

 the factors of f ' x, or of the series 



1 - SAaX^ + 5A,x*- 7A,x« + &c. 

 But it includes all the factors of \I/x; and since 4)'x and \I/ x 

 are similar functions, it now follows that if they differ in any 

 respect, 3 A3 will differ from A,. An inspection of the known 

 composition of these quantities, however, instantly shows us 

 that they are equal ; consequently 



ip' X = ij/ X 

 and by applying this to relation, ( 1 ), we likewise obtain 

 •l)' X = — <^ X 

 These two relations determine all the derivatives of ^ and 

 4/, and if x is put = in developments similar to development 

 (2), we immediately obtain the simplified series 



^^ = ^ - 1-5-3 + 1-2-374- - ^"- - 



x' X« o 



^^='-1^2 + 1.2.3.4.5 -^^•- 

 whence the important forms 



(♦x or si/i X = = 



x\/^i , -"VITi 



, t + e 



\j/ X or cos X = 



