434 Mr. J. H. Michell on the 



and therefore 



^=fsini0{(l-&i + 4V-&i 8 -2M 8 )cos0- 



+ (5& 1 -2& l 2 -4& 2 + 106 1 & 2 + 56 1 3 )cos3«/) 



+ (4& 1 2 + 85 2 -7& 1 ?> 2 -^3)cos50 

 + (1 16i b 2 + 116 3 ) cos 70 \ . 

 To put -=^ into a similar form we use the Fourier expansion 



sin ^27+10— gj = SA 2n+ i cos (2n + 1)0, 



where 



6V3 6r + l 



A 2n+1 _ ^ 32 (2n + l)2_(6r+l) 2 



Since 





-£ = -sin*0^sin J0- ^ + ^sin 20 + -J0- ^ + . . .1, 

 we get 



-p = Sin* 0[-J- COS + gV C0S 30 + 224COS 50 + lio cos ^0 + • 



+ &i(— /q cos + ^ cos 30 + jf^ cos 50 + ^^2 cos 70+ . 



+ & 2 ( — T6 3 C0S — 41 C0S ^0 + 5 6 COS 50 + 2% C0S 7 $ + • 

 + h{ ~ 3% C0S <£ ~ 2*80 C0S 3< £ -t¥(J C0S 5 + 8 C0S 7 </> + • 



+ ...]• 



Equating the coefficients of corresponding cosines and writing 

 18 y/3gjir=k, the following equations are obtained for k and 

 the 6's ":— 



l-^ + 4V-V-2M 2 



= /H-125~-1756 1 --081256 2 --053986 3 ), 



5ftj _ 26 x 2 - 46 2 + 105A + 5^! 3 



= /:(-0125 + -2l8755 1 --14772^ 2 --067865 3 ), 



4^ + 8^-76^ -76 3 



= ^(-00446 + -0397736 1 + -2321436 2 --139706 3 ), 



11&& + 11&3 



= £(•00227 + -01785*! + -04779& 2 + -23756 3 ). 



