562 Proceedings of the Royal Irish Academy. 



~ 2» - 2M 



Vhaf a\P 2) -^ '^^^[2a da +2 da^ )_ 



_{a+8ay ft^V 2 

 J/^ftS cos(2Z7- 2r) 



/ 27J dA"^'\ 



+ 2?h' , ^'*' + « -- — cos (ko - Z-i;'). 



\n -IV da j 



To solve this, assume 



{a + S«)8/- = P cos (2 CT- 2v) + ^Q^o'*^ cos (^-y - W), 

 and put 



whence, by substitution and identifying the coefficient of cos (2 TJ - 2v) 

 with zero, 



-p(2;i/-2.r+Piy^-|i^jf^«- 



whence 



Jf2 6« - 3 J/ , 



= i\^2 _ (2J/-_ 2?0' 2?i - 2i/"'^^' ' 



J/ 



but iVsii ?i and — =0= ; 



n 



M- 

 whence P ^^ 5- «^. 



Identifying the coefficient of cos {hv - kv') to zero, 



- Qu o'*) (X7J - knj + N^ Q„ ^'-"^ + m' ; A-'^ + ^ __ ) = 0, 



\n- n da j 



{kn-kn)--j\- \?i-7i da J 



whence, finally, we have for 8r (omitting Ba . Sr), 



TJ2 



8r = - rt — co^(2U-2v) 



