1893.] On Operators in Physical Mathematics. 107 



powers in A are zero, if we follow the law of the coefficients. So A 

 is also complete, and C must be some combination of the series A 

 and B. In fact, if we assume that 



u = 



is a solution of the characteristic (4), and insert it therein, to find 

 the law of the coefficients in the usual manner, we find that the even 

 i's are zero, whilst the even a's are connected in one way, and the odd 

 's and even &'s are independently connected in another way. This 

 makes 



w = aA + &B, (8) 



where a and 6 are independent multipliers. Now, judging from 

 common experience with this rule-of -thumb method of constructing 

 solutions of differential equations, we might hastily conclude that A 

 and B represented the two independent solutions of the characteristic. 

 Here, however, we know (analytically) that they are not independent, 

 but are equivalent. Therefore 



C = aA + 6B, (9) 



where the sum of a and b is unity. It only remains to find the value 

 of a. This is easily obtainable, because the separate series in (6) are 

 rapidly convergent. But we need only employ the first series, viz., 

 to find the coefficient of x. Thus, the first line of (6) gives 



v/2 



1-1106 



We see, therefore, that the series is algebraically identical with 

 half the sum of the series A and B. 



To further verify, we see that the coefficient of x in (6) should be 

 2/7T times that of a?. This requires 



or 07068 = 0'7067, 



which is also close. Similarly, from the x* series we require 



or 0-277 = 0-277. 



