Mr. 0. Hea-viside. [June 15, 



= (C + log A->*-' + (A-A 2 |H- A 3 |2- A 4 |s + . . . . ) 



...., - (M7) 



Avhere we see that the operator (143) appears. Integrating, we 

 have 



)IW, (148) 

 comparing which with (140), we see that 



; (149) 



for which a verification would be desirable. 



Companion FormuJce, H (g^) and K (g;r), derived- from Companion 

 Operators, expressed in Descending Series. Also in Ascending 

 Series. 



68. Passing, however, at present to more manageable operators 

 involving the two solutions and different forms thereof, it will be 

 convenient to introduce a notation and standardization which shall 

 exhibit the symmetry of relations most clearly. Thus, let 



vu**+ <*&*** aw) 



Here q is a constant and v is d/dx. Superficially considered, these 

 functions only differ in one being i times the other. But the common 

 theory of the imaginary does not hold good here, or in operators 

 generally. In a descending series we have 



(152) 



as already shown. It is twice the function C, equation (3). Simi 

 larly, we may integrate (151). Introduce the factor e~?*, thus, 



t 



-V 2 )* V-2 



