578 T. H. Havelock. 
with 2 a positive integer. The cases which occur in (19) could be tabulated 
directly by means of convergent series and asymptotic series. They can, 
however, be derived by repeated integration of the second Bessel function, and 
can be expressed in terms of functions of which Tables are now available. 
The functions satisfy the relations 
Ie == 1 ny 
PD: =D a) G@al) Bn (22) 
Further, in reducing a function of positive order n by this relation, (22) 
holds as far as 
2P2 = p(Pi+ P-1)—Po, 
P, = p(Po+ P-2). (23) 
Now we have 
Re [esi (p seep) dp = —7 ie (pyle (24) 
where Yo is the second Bessel function defined by 
Y= = ine sec d cos (p sec h) dp. (25) 
We shall use the notation 
Yoo= [Yo (avd. (26) 
Since we have 
poss —F Yo; P2=—2V! = > Va (27) 
it follows that by using (22) and (23), we may express the unknown integrals 
in (19) in terms of Yo}, Yo and Yj. 
Some numerical values of Yo! have been published recently by G. N. 
Watson; these are not sufficient for our purpose, but Watson also gives 
Tables of Struve’s functions Hp and Hy ranging from 0 to 16. In terms of 
these functions 
Yoou= pYo-F p (YoHi—YiHp). (28)* 
Watson’s Tables of Struve’s functions and of Yop and Y, have been used in 
the calculations that follow. 
6. Returning to (19) we evaluate the simple integrals and reduce the 
others in the manner indicated in the previous section ; omitting the algebraic 
reductions, the final result is 
* G. N. Watson, ‘ Treatise on Bessel Functions,’ p. 752 (1923). 
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