352 Mr MURPHY'S THIRD MEMOIR ON THE 



— , hence the quantity we are considering must be zero, and there- 

 fore 



2w + 3 X {2n + 2) (2w + 5) x(x—l) 



~ i '2ra + a; + 2'*' 1.2 ' (2n + x + 2) {2n + x + 3) 



_ (2» + 2) (2?^ + 3) i2n + 7) x{x-l) {x-2) „ 



~ 1.2.3 ' {2n + x + 2){2n + x + 3){2n + x + 4>) "' 



which shews the generality of the observed law of the coefficients ^„+i, 



Substituting now these values in the general formula for V,„ we 

 get the required function which is reciprocal to t", namely, 



rr- r. /„ „v T, (2« + 2)(2w + 5) „ 

 K = P„+{2n + 3).F„^, + ^ ^ ^ . P,+8 



(2w + 2) (2w + 3) (2w + 7) „ „ 

 "- 17273 •^"*^' *'''• 



25. The Junction which is reciprocal to t" is transient. 

 For in general 



d".(tt'Y _ d^itt'T 



\.2.3...ndt'' ^ ^ ' 1.2.3. ..ndt'"' 



and putting 1-t' for t, and expanding the binomial (l-t')", and 

 lastly actually performing the differentiations indicated, we have 



^ A) .^n-A 11*^1.2 1.2 



and therefore 



V-l) ''n-JA ii'^+i2 1.2 * 



-(2w + 3)|i T" • ~T~ ~~T72~~ • 1T2 ' 



(2«+2) (2W+5) ,, »+2 n+3 - (w+2)_(^+l) (w+3) (w+4) ,,2_^ > 

 + 172 t^~ 11 1.2 • 1.2 ■ "^ 



-&c. &c. 



