140 Mr. 0. Heaviside. [June 







of KoCg^), and two of J (s;r), in obtaining and harmonizing the p 

 vious three forms. It would therefore appear probable that there is 

 an additional principal formula for K (<2#), and another for J (s), not 

 yet investigated. 



Conjugate Property of Companion Functions. 



71. The conjugate property of the oscillating functions is 



JoOwO ^ G (*aO -G (^) ~ Jo(^) = ~ , (170) 



using the pair (162), (163), or the pair (158), (161). And, similarly, 



H ( 2 oO ^K (^)-K (^) ^H.( aa! ) = - . (171)- 



But, in the transition from (171) to (170) by the relation q = si, it is 

 indifferent whether we take H (qx) = 2l (qx) = 2j (s#) or e ^ se = 

 3o(sx) iG (sx'). This conjugate property is of some importance in 

 the treatment of cylindrical problems by the operators. 



Operators with two Differentiators leading to H and KO and showing 

 their Mutual Connections compactly in reference to Cylindrical 

 Waves. 



72. The fundamental mutual relations of H and K are exhibited 

 concisely in the following, employing operators containing two 

 differentiators, say v an d <?, viz., 



- (1?2) 



Here it should be understood that either v or q may be passive, when 

 it may be regarded as a constant. But when both are active, there 

 are two independent operands, one for v & n d the other for q. In a 

 cylinder problem relating to elastic waves, we may regard v as being 

 djdr, where r is distance from the axis, and q as d/d(vt), where t is the 

 time, and v the speed of propagation. We have 



