3SS Mr MURPHY'S THIRD MEMOIR ON THE 



the reduction of which to the powers of t is effected as before, putting 

 — m for m, whence 



(w + ot) (w + m-l)...(l +»?) ( n n + 2m + l 

 ^"~ 1.2...ra ^ 1* 1+m ■ 





When j» = 0, 



« o 1 ** " + i M«:il) (w + i)(w + 2) 



which is the same as the value of P„, Sect. ii. Art. 2. 

 When m= — ^ 



2 



and t = sin^ ^ 



2 



{(w + ly-l^{(>^ + ly-2-} _,^.^,0 



2.3.4.5 



Q»=2.4 2n •il-1.2'^ ''" 2 + 1.2.3.4 -^ ''" 2 *'''-^- 



14. To express the quantities Q„, q„ by means of a differential 

 equation. 



Suppose /{t) is a function of #, subject to the condition 



t(l-i) ./"it) + (»»+ 1) (1-20 ./' (0 + « • (« + 2/»+ 1) ./{t) = 0, 



where /"(^ denotes the second, and /'(^) the first differential co- 

 efficient of f{t) relatively to t ; differentiating this equation, we get 



t{l-t) .f"{t) + (»« + 2) (1 - 2^ ./"(/) + (»- 1) (« + 2»? + 2) ./' (^) = 0, 



^ (1 -0 ./"" {t) + (w + 3) (1 - 20 ./'" (0 + (« - 2) (W + 2»« + 3) ./" {t) = 0, 



and generally, 



/(i-0/""^'no+('»+^-i)(i-20-/""'^'"'MO+(^-^+2)(«+2/»+r-i)./"'<'-»(0=o. 



