258 PROFESSOR POWELL'S REMARKS ON THE 



|3 COS h I 2 sin^ ( — £— ) sin {nt ^hx) — sin A: A ^ cos {nt ^ k3c)\ 

 A^= - 



+ 13 sin 5 I sin k ^xmn{nt — hx) — 2 sin^ (— ^) cos {nt — hx)j. 



Also, differentiating them, we find. 



(17.) 



^-p = -— ^2 05 gin (72 ^ — ^ 0?) 



(18.) 



<p; 



> 



(20.) 



^ = — w2(3cos^sin (w^ — Arzr) — w2|3siiiicos(w^ — A;.r) . . . (19.) 



On arranging the terms in (14.) we may write for abridgement, 



jp=:9(r) H-'^/Cr) A3/2 



y = (p (r) + •+ ir) A ^2 

 y = -v^/ (r) A 3/ A 2; 



and those forms (14.) will thus become, 



^,= 2[pA, + yAg (21.) 



^U^O'A^ + yA,] (22.) 



On substituting in these forms the values of A ;; and A ^ above, viz. (16.) (17)? we 

 shall have, 



g'{/3sin5} sin 2 ^ 



+ y {/3 cos^' 



— P {a 



— q {^ cos h 

 .— q {^ sin h} 2 sin2 6 J 



jo' {j3 sin &} sin 2 ^ -1 



y sin (w ^ — A* a:) "1 





2 sin2 ^ 

 - sin 2 ^ 



► sin (n t — k x) 



cos (nt — kx) 



(23.) 





-\- p' {^cosb 



■2 sin2 d 



J j2 -<* A 



Isin 2 ^1 , ^ ^ 



' J > cos (w ^ — « 0?) J 



(24.) 



+ q {c6 



— q {cc 



— p' {jScos^^ 

 L-y {|3sin^>}2sin2^ J 



On comparing these expressions with those for the same functions (18.) (19.) which 

 must be identical, and equating in each the terms which form the coefficients respect- 

 ively of sin {nt — kx) and of cos (nt — kx) (since they must hold good for all 



I 



