Manchester JMevioirs, Vol. i. {igo6), No. ^. ii 



- 2J^{rE^){JlrE;) + J.XrE,) + J,{rE,)}cos <p 

 + {2JJrE^)[JXrE.^ + J.^rE.^] - J^irEy^rE.^) 



- J ^(rE.) J ^{?Ei)} cos 2(j> 

 + 2J,{rE,){J,{rE.^ - J,{rE,)}cos s<p 



+ 2\J,{rE;) + JXrEyirE.:) ^ J .{rE^J.irE,) 



-Ji{rE^)]\ cos 40 

 + 2[JJ,rEyi,rE,)+J^{rE.:)[JlrE;) + J^{rE.)]\ COS 50 



+ 2 \J,{rE;) + JlrE.) + J.^rEyirE.^ + JlrE^^rE;^ 



+ J,{rEi)J^{rE,) + J,{rE,)J,(rE,)] cos 60 . . (15), 

 and 



sinrljE*! sin + ^2 sin 20 + ... + ^esin 60} 



= 2{JSrEy,{^E^ + jpEyirE.;)} sin 



+ 2{ J.(^^2)K(^^i) - Mr^i) - MrEy)J,{rE,)] 



+ J.lrEyj^rEi)]%\n 20 



+ 2{ j;(r^J - Jl^rEyirE.^) + .U.Ey^{rE.)] sin 30 



+ 2{J^{rEXJ.^rE^) - J,{rEy{rE.)] + J,lrEy^{rE,) }sin 40 



+ 2{ J^l^^i) + -/i(^^5) + J,{rE,).h{rE,) 



+ JlrEypE:)]smsf 



+ 2{ J-:(r^6) + ./3(^^2) + J,{rE,)[.h{rE,) + .7i(r^i),/i(r^3)] 



+ J2(''^i)«A(''-£"4)} sin6(/i . . (16) 



The coefficients A^, y?,... of the expansion of in (8) 

 can now be written down. They are 

 A,= - 2J,{E„){J,{E,) + J.XE;) + J,{E^,)} 

 -2{J,{Ey,{E.) + J^Ey,{E,)}, 



2A,= 2{/,(2^,')[/„(2E.,) + Jl2E.^] - Jl2Eyi2E.^ 



-J,{2E,)J,{2E,)] 



- 2{ Ji(2^,)[./„{2^l) - JI2E,) - J,{2Eyi2E,)] 



+ J,{2Ey,{2E,)] 



2,A.= 2J^{2,Ey,U^Eyj,{2>E:)\ 



- 2[Jl2>E,) + Jo{2,Ey.{zE,) - J,{zEyizE.y, 



