204 Mr. W. Steadman Aldis. 



When n is a positive integer, a second solution is given by 



y = VA n (x) (4), 



whyre 



1 . n - I 1 . 2 . n - 1 n - 2 



2n-2 



* 



2, 



where S r denotes the series 1 + - + - + ...... + - . It is also necessary to 



23 r 



assume zero as the value of the, in itself meaningless, symbol S . 

 It is also known that if E represent the quantity 



the function EI n (x') - A n (x) becomes indefinitely small as x increases 

 indefinitely. Hence since in virtue of (3) and (4) 



y = B{EI(a;)-A n (aj)} 

 is obviously a solution of (1), it is allowable to write 



K n (x) = EI n (x)-A n (x) ..................... (6). 



3. The three functions I, A, and K are all subject to the laws 



where- Ivi is written for I n (%)- These equations hold when either A or 

 K is substituted for I. When n has the value zero, the two equations 

 must be replaced by the single equation 



~dx ~~~~ I i > 

 or the same with A or K written for I. 



