J.] On Operators in Physical Mathematics 



> 

 - - qI (qr), [a] 



/_V_V _ ! TT r & -) 



141 



(173) 



..[d] 



- , [e] 



where the letters in square brackets are for the purpose of concise 

 reference. Similarly, we have this other set, 



V<7 T / , \ r * -i 1 



.... 7-5 jrr = vJ-oU^vjj L-^J 



/ ft* TT-MS ' u 



, -[B] 



(174) 



= i,..[D] 



/ 2/2 2\4 ' L J 



TTl ?.' C ? ) 



1 



The first set is usually, though not essentially, concerned with an 

 inward-going, and the second set with an outward-going wave. The 

 exchange of r and vt and of v an d <?> transforms one set to the other, 

 so that the proof of one set proves the other. 



In obtaining [a] from [P] we regard q as a constant, or at any 

 rate, as passive for the time, expand [P] in descending powers of v> 

 and integrate directly with the result [a], as in 13, equations 

 (28), (29). 



To obtain [6], introduce the factor e? r to [P], and expand the 

 transformed operator in descending powers of q, as in 14, equations 

 (30), (31). 



To obtain [c], we make q passive, and introduce the factor e~ vt v . 

 Then expand the transformed operator in descending powers of y> 

 and integrate as in 68, eqautions (153), (155) (only there the 

 operator is q, making the case [C]). 



