918 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



In these, as in (11), p>p' and z>z' ; for p<p' or z<.z, interchange p, p or z, z'. 

 The definitions of the coefiicients which occur in (11), (13), (15) are, in the notation 

 of Art. 3, 



B<(,, VI = «!, mBi, „, + ttj, ^^2, ,n + 0-3, m^.\. m 



Ag,™ = /?!, „.Ai, „. + ^o, ,„ A2, ,„ + ySg, ,„A3, „. 



Be,™ = ^i,», Bi_ ,„ + /Jj, ,„B2, ,„ + ^3, ,„B3, ,„ 



Ce.Dl = ^1. „,Ci, ,„ + ^5, ,„C2, m + A. »i*-'3, ni . 



A^,m = yi.mAi,,,, + y2,,»A2,„ + y3,,„A3,,„" 

 B^,i)i = yi, ,„Bi,m + y-i, mBj, ,„ + y3, ,„B3,, 

 C^,,,i = yi, „iCi, ,„ + y2, ,„C2, m + y-i, ,„C3^ ,„ 



(16) 



For the case m=0, the form of the source functions is slightly modified, 15 (3), 

 but the method remains the same. The form of the results can be simplified as in 

 Art. 9. It will be sufiicient, however, to state concisely that balanced solutions for 

 the sources 



(<^,^,,/.)=DrV-i 



are obtained by taking 2' of (11), (13), and (15). 



m 



17. The loermanent terms in the solutions for auxiliary sources 



The permanent terms in (11), (13), and (15) of the preceding article can be 

 determined by straightforward algebraical expansion. The work is easy, if somewhat 

 tedious, and only the results will be given here. 



Solution for source <p = D~^r~^. Permanent terms. p'>p', z^z'. 



. 2v(v - 2) _., j 1 . ,^3 1. ,, , I lv2-2v+16, 'X , 1 '2 -iKi'-S)/ n 

 0= ^ 'a - < ^(z -zY - -u- z)p- > (z- z) + -p^a ^^- — '{z -z), 



with 



(1) 



cf) = p\z - z')p 1 COS (o) - w' 



+ a" V 

 1 



"^Z { |(^ - '^yp - i(^ - ^>' } + <v^ - ^)p^ 



cos (a> - w') 



8 



a-^p 



a-ip'iAf^(z - z')p cos (o) - w'). 



^?i I y(^ - ^')V - |(^ - ^>^ } + B^;_\(. - z)pa'\ cos (a, - o,') 

 -la-VXV^-2>cos(co-co'), 

 ^= «"V[CJ',\{^(^-^?P-|(^-2V} +Cj'_\(.-.>a^]sm(cu-(o') 

 -la-V^C^>-^')psin(a>-o>'), 



(2) 



' I 



and 



z 



t^ = '-p'"'{z - Z )p~"' cos ??z(a) - oj') + «"-'>""A^"^„^(2 - ^')p"' COS m(to - co'), 



>(0) 



a-'-y^^qr' ^J^z - z')p"' cos ?n(w - w') 

 «- V"Cj;\,(^ - 2')p"' sin m(a; - a,') ; j 



(3) 



