1893.] On Operators in Physical Mathematics. 135 



When n is not an integer, !(#) and !_(#) are different, and repre- 

 sent two independent solutions of the characteristic differential equa- 

 tion. But when n is any integer, positive or negative, they become 

 identical, so only one solution is got. Then another is (when n = 0) 

 represented by the rate of variation of I M (#) with n when n = 0. 

 Thus, 



i , . , 



2 ]2<l + w.)(2 + ) J 



which, when n = 0, is by inspection the function on the right side of 

 (140). Notice that this method of obtaining the second solution, 

 like the just preceding method, gives it immediately in the form 

 properly standardized so as to vanish at infinity. The constant C 

 comes in automatically, and requires no separate evaluation. 



The Operator producing K (#). 



66. But our immediate object of attention should be the function 

 on the left side of (140). How it can be equivalent to the right 

 member is a mystery. It is certainly an extreme form, if correct. 

 We may write it in the form 



A-A 2 |l + A 3 J2-A 4 |3 + . . . . , (143) 



where A is d/dy. Now the other function IQ(^) is 



without any mystery, and we see at once that these forms are ana- 

 logous to 



e-*= A A 2 + A 3 -A 4 + ____ 9 (145) 



e* = l + A- 1 + A- 2 + A- 3 +.... , (146) 



the latter, corresponding to (144), being obvious, whilst the former, 

 analogous to (143), is an extreme form already considered and ex- 

 plained ; see equations (71), (72). The unintelligibility of (143) is 110 

 evidence of its inaccuracy. More puzzling things than it have been 

 cleared up. 



67. We may also employ the special formula (126), of which we 

 had separate verifications. Multiply it by e* and then write A" 1 for 

 x. Thus, 



