330 Mr MURPHY'S THIRD MEMOIR ON THE 



Ex. 2. Given fi(p(t) .cos {at) = cos (a 0). 



As before (p{t) = c„ + Cj cos t + c^ cos 2t + CaCosSt + &c. 



sin flTT fl 2«cos^ 2« cos 2^ 2a cos 3^ „ ] 



cos at = < J \ h &c.> 



therefore cos a0 = sin(a7r)l- - -~- + f^' „ ^^j + &c.l 



' [a a^ — 1 a^ — 2' a- — 3^ J 



But also by reciprocal functions we get 



sinaTrQ 2acos0 2acos20 2acos30 „ 1 

 cosae = __ |- _ _,__ + --^-^^ -,__ + &c.} 



TT 1 2COS0 2cos20 2cos30 „ 

 Hence Co = - , c, = , c, = , Cs — , &c. 



TT TT TT TT 



therefore 7r^(#) = 1 +2cos0 cos^ + 2 cos20 cos 2^ + 2 cos36 cos 3# + &c. 



or 27r(pt= 1 +2COS {9 + t) + 2 cos 2{e + 1) + 2COS 3{9 + 1) + Sic. 



+ l + 2cos{9-t) + 2cos2{9-t) + 2 cos 3{9-t) + &c. 



^ (1-A)(1+^) (1-A)(1+^) 



l-2h cos{9 + + A' 1 - 2A cos (0 - ^ + *' 



when A is put equal to unity. 



Ex. 3. Given ft <{> (t) : cos {at) = 27'(a). 



jP(a) must be such (in continuous functions) as not to change when 

 — a is put for a, since cos (at) which is under the sign of integration 

 will not then alter its value. 



Proceeding as in the former examples we get 



ET/ ^ • \<^o ac, ac, acs „ ] 



F{a) = sm«. |- _ -,_^ + -,_^ _ -^-^^ + &c.} 



Put successively a = 0, 1, 2, 3, &c. hence 



Co = - . 1^(0) , c. = - . F{1), c, = -. F{2), &c. 



ir IT TT 



hence 7r(p{t) = F(0) +2F(1). cos #+ 2F(2) . cos 2# + 2F(3) cos(30 + &c. 



