356 Mr MURPHY'S THIRD MEMOIR ON THE 



h being put = 1, after the differentiations; this value of 1.2,..?iF^, is 

 expressed in two finite series, each containing only w + 1 terms. 



If we actually add the terms in this formula, which contain the 

 same powers of A, we get 



V - 1 K:? w + 2 h d"^^R (n + 3) 



"~ 1 . 2...W \dh" "^ w + 1 ■ 1 ' dh"^' "^ (w + 1) (« 



, n 



h' d'^'R 



+ 



n + 2)' 1.2" c?A"+' 



(« + 4).ra(w-l) k" d'+^E 



^ SL 1 



{n + l){n + 2){n + 3)'l.2.3' dh 

 when h is put equal to unity. 



27. -Discussion of the transient function N ^. 



Put « = in the general expression for ^„ in the preceding article, 



dR 



hence V. = R -^ 2h -y^- 



dh 



= U-2A(l-20+A^}-^ + 2A(l-2jf-A) {l-2A(l-2^) + A^}-i 

 (1-A)(1+A) , , . 



This function, as has been observed in Section vii. (22), is in general 

 zero, except in the particular case when ^ = 0, when its value is infinite. 



If we imagine a curve of which the equation is 



y {l-2A(l-2a;) + ^^}4' 



where h is less than unity but nearly equal to it, the limiting values of 

 y as A approaches unity, will give the geometrical interpretation of 

 the transient function V^. 



