138 Mr. 0. Heaviside. [June 



where J (stf) is the same as in (158), and G (*#) is its oscillatory coi 

 panion given by* 



(161) 



What is obscure here is the getting of only one oscillating function 

 from I (^a;),and of two from K (qx). In corresponding forms of the 

 first and second solutions we should expect both oscillating solutions 

 to arise in both cases. However this be, the transformation (160) is 

 in agreement with the other form (155). For, if we make the change 

 q = si in it, we obtain the same formula (160), provided J and G are 

 given by 



J (s#) = I ) |E (cos + sin) sx H- Si (sin - cos) e j , (162) 

 G O (B) = ( ) R(cos sin) sx + Si (cos + sin) s# , (163) 



\7TSXj [_ J 



where E and Si are the real functions of sx given by 



1 2 3 2 5 2 



(164 



+ .. . = x ( ' LOU I . ,, 



E = i+ -"+ -^+.... =l-^-r+ -...., 



~ Scjaj J3(8go?) 8 " i \8sx _ 



Now here- (162) is Stokes's formula for J (aj), known to be equi- 

 valent to (158). And (163) shows that this kind of formula for the 

 oscillating functions allows us to obtain the second solution from the 

 first by the change of sin to cos and cos to sin. The function 

 GO(SX) of (163) may be shown to be equivalent to the G (sx) of (161) 

 by other means, and certainly verifications are desirable, because 

 transformations involving the square root of the imaginary are some- 

 times treacherous. 



Transformation from H (ga.') to the same J (.?) and G (sx). Explana- 

 tion of Apparent Discrepancies. 



70. Now as regards the changed form of the Ho(jaj) function of 

 (152), there is a real and once apparently insurmountable difficulty. 



* I have changed the sign of K and Gr from that uped in my 'Electrical 

 Papers ' (in particular, vol. 2, p. 445), in order to make them positive at the 

 origin. 



