1893.] On Operators in Physical Mathematics. 139 



We know that H (<p) and 2l (qx) are equivalent, both analytically and 

 numerically. Why, then, does the first become complex, whilst the 

 second remains real when we take q = si ? They cannot be both true 

 in changed form. Thus (152) becomes (doing it in detail) 





= ( - ) R, (cos -I- sin) s# + Si (sin cos)s# 



\7TSXJ L J 



/ ) R(cos sin)s-f Si (cos + sin) so; . (166) 



\7TSXj l_ J 



That is, using the functions (162), (163) again, we have the trans- 

 formation 



= J O (B) -G (B), (167) 



whereas 2T (2^) becomes 2J (saO. This was formerly a perfect 

 mystery, indicative of an imperfection in the theory of the Bessel 

 functions. But the reader who has gone through Part 1 and 27, 28 

 of Part II will have little trouble in understanding the meaning of 

 (167). The functions H and 2I , though equivalent (with positive 

 argument), are not algebraically identical. To have identity we 

 require to use a second equivalent form, so that, as in 28, 



In this form we may take q = si, and still have agreement in the 

 changed form. We obtain the relation (167), provided that 



1 1 2 3 2 1 2 3 2 5 2 sV s*x* 



(169) 



A.S I mentioned before in 22, this formula for G (s^) may be de- 

 duced from formula? in Lord Rayleigh's ' Sound,' derived by a method 

 due to Lipschitz, which investigation, however, I find it rather 

 difficult to follow. 



We have, therefore, three principal forms of the first solution with 

 q real and positive, viz., I (qx), ^'H (qx^ and the intermediate form 

 (168). We have also three forms of the oscillatory function G (sx), 

 viz., (161), (163), and (169). But we have only employed two forms 



