130 Mr MURPHY'S SECOND MEMOIR ON THE 



and putting x = 0, 1, 2, &c. successively, it follows that fu"U and its (« — 1) 

 successive differential coefficient vanish when u = and w = - oo , Hence 

 U=0 has n real negative roots; and therefore X„ = has n real positive 

 and fractional roots. 



15. In general let U,„ V„ be any functions of the variable t and the 

 integer n, and let A-^.-.A,,, ai...a„ represent constant quantities; or de- 

 pending on n only. 



Put T„ = C/„ + A,U, + A,U, + .... + A^U„, 

 and T:= K + «i^> + «-.F, + .... + a,r„. 

 Then the n equations 



j;r„r„=o, f,T„r,=o, j,t„v,=q ;r„r;_,=o, 



Avill serve to determine the constants A^, A.,....A„. 



In like manner let the corresponding constants «i, a2....a„ be de- 

 termined from the n equations 



the functions T„ and T„' which are thus determined, are reciprocal func- 

 tions, and possess the general property ft T„ TJ = 0, except when n - n', 

 and then 



ft 2\ T: = aJtT^K = A,, ft T: C7„ ; 



this is the general principle of reciprocal functions. 



Cor. Let f{t) be any function of t represented by the series 

 f{t) = c,T, + c. 2\ + c, T, .... &c. 



where Co, c,, Cg, &c. are constant coefficients to be determined, then 

 multiply by T^, T(, T~U &c. and integrate the successive products, 

 and we get 



c,ftT,Tl = ff{t)T^, 



c^ftT.TI^ ftf(f).Tl, 



c.fT,T^ = f,f{t).T.I, 



&c &c. 



by means of which equations the required coefficients are given. 



