492 Mr. Moon on the Symbols sin a> and cos <», 



=: —e~^^ {cos ri/— \/ — l sinrj/) x 



' j/y(cosQ— \/^sin5) jo?/^~^(cos2g- V'^sin2g) 



. / i\ „_2(cos3e— 'Z — IsinSfl) 

 + iJ(p-l)j/^ ^^ 7^ 



+ &c. 



^ ;?(jp— l)...3.2.1(cos7)+1.9- -/ — Isinjj+lJ) 

 j-p+i 5 



therefore, equating together the possible and impossible parts 

 of the above expression, and observing that 



(cos ri/— \/ — \ sin ry) (cos fl — v^ — 1 sin fl) 



we find 



/ yPs-qy Qo^ry=i 



= cos {ry + 9) — V— 1 sin (rj/+ 6), 



/ 



cos(rj/+fl) + 



p-y 



p-\ 



r 



-cos(;7/ + 2fi) 



-s-^y^ 



f- 



+ ^^%-^^^~^ cos {ry + 3 ^) 



+ &C. 



p.(p— 1)...3.2.1 



+ ^+1 *^°^ (^3/ +P + 1 .^) 



' ^'^'"j:^^ + ^^ +^-^'cos(r^+e) 



-s-!?y^ + ^ yr^V "^sin(rj/ + 2fl) 



+ &c. 



2?.f»—l)... 3.2.1 . , 



+ ^ ^^^^j, sin(n/+;.+ l.e) 



formulae which hold for all values of j/, provided only the 

 original conditions upon which they were deduced be still 

 maintained, — 1, that p be zero or a positive integer; and 2, 

 that either q ox r he finite, and not otherwise. 



If q be finite and positive, every term of the above expres- 

 sions vanishes when j/ = 00, and it is easy to see that the values 



/ yP e~^V cosry and / yP s~9y s'lnry 



(B.) 



