920 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



and, for ^n>l, 

 A?>, 



2(7/1+1), 



^^^u; ^ m-1 ^ 2 



l(0) 



e.m 



m m[m + 2) ' 



B«. -icn^m,,*,), if,= """'""''^ -^-il^). 



7)1 



C» = -2(„..1X,„..), e= -=!±>_(™zl)(51zi.+ '),_|z^. ^ 



?« 



m 



(10) 



Solution for sowce ^ = D/^r \ Permanent terms. p'^p',z^z. 



,= -(.-.)IogA_iC.-.{(i^-(^-f)^^.(.-.)^[-^„-.{j^^ 



(11) 



ivith 



4> = a"" V 



^. { |(^ - ^')V - |(^ - Op^ } + A^\(. - .')P«^" 



-1«-V='A^^\(. -.>(-) sin (o 



a ^p 



)f 1 1 |(^ - ^'fP - i(^ - ^^)P' } + C('^ - ^>«' 



( - )sin (to - oj') 



( — ) sin (co — oj') 



- 1«-VX\(2 -.>(-) sin (<o - o,'), 

 i// = p'(z - //)p~^ cos (o) - Hi) 



+ a" V 



C 



.(2) j 1 



i|4(^-^?p-|(^-^v}+cJ>-.> 



cos (w - oj') 



- -o-« V'^^ Ji(?- - Z')P COS ((O - (o'), 



(12) 



anc^ 



/ 



<f) = a -'" 7/'" 

 1 



2 



A^!m { -g-(^ - ^')'>"' - ^^ J^ ^^ (g - 2')p'"^^ I + A|;|,„/;j - r/)p"'a^ ( - ) sin m{^ - «') " 

 e = a-'"- V"{l>;^|„ { \{z - .')V" - ^J-^^jz - z)p- -^ \ + B';^Jz - ;')p'"«-']( - ) sn. m{o. - o/) 



if/ = —p'"'(z - z')p~™ COS ?rt((/J — w') 



(13) 



c'f' i A(z - ?/)>'" - -^ , J- -- (3 - /)p"'+^ I + c7 (z - /)p"tf 



cos ^y^(oJ - o)') 



\ 



where 



4(,^4. 1)"~"""V""''G^_„(2 - ?/)p'" cos »<co - 0,'), 



Km+1) 





^^(0) ^£ J_ 2 7v-64 



"^.i 3 8-v ^ 3 (8-v)2' 



,,(2) _ _ 8(tH- 6) pj(.i) ^ 2^ vjMO \ (v + 6)(7v-64) 



*•! 8-F ' "^'i ^3 8-v ^ 3 (8-.')''^ 



80 



(,(2) _ 



^.i~ 8-v' 





v-\1 10 7./ -64 



V ~ 3 (8 - v)2 



(14) 



