DIFFERENTIAL AND INTEGRAL CALCULUS. 49 



the n h differential coefficient of (x*u) is 



jn J n ~l J n ~Z 



yd u ct u f ct u 



for the like reason. In general, if one of the functions be 

 a rational algebraical function of r dimensions, the ?i th 

 differential coefficient (n>r) will only contain (r + 1) terms, 

 since the (r + l) th and all subsequent differential coefficients 

 of this function will vanish. 



The n* differential coefficient of 



is 



~ Nra+1 , and of 7- is 



\ 1-1 ] n _\_ h.'Yt 



a- ar ~ (a - aO"* 3 a"4 to (a + ^) n+] 



We can find the n differential coefficient of , - by 



x z - (? J 



separating it into the sum of two fractions, with denomi- 

 nators x c, x -f c, and so with any algebraical fraction 

 whose denominator is the product of simple factors; of 

 course all are so, but if the factors be unreal, we shall have 

 to use De Moivre's Theorem to get the result in a real 



form. Thus, if y - , the factors of x 2 x + 1 

 are x a, x /3, where 



and if we assume y= -+- 3 so that 

 x a. x- 13 



II 



