132 Mr MURPHYs SECOND MEMOIR ON THE 



17. If it is necessary that the terms which compose the reciprocal 

 functions S,„ S,! should follow a simple law, it will be most convenient 

 to get first two reciprocal functions of t, as R,,, R,', which may contain 

 an arbitrary constant r, and to put for «„, J„, c„ &c. the values acquired 

 by R„ when r = 0, 1, 2, &c. ; and similarly for «„', i„', c,,', he. the cor- 

 responding values of R\. 



Example : 



Thus, put R^^iiff'"^-^, and RJ = {ttf'" '^, P„ being the 



d" (tt'Y 

 function so denominated in Art. 3, namely, — ^,^ ; then, integrating 



by parts, we have 



the part outside the sign of integration vanishes between the limits of 

 /, and repeating the same operation any number of times, the part out- 

 side the sign of integration is evidently of the form 



dt'-" ' dt"-' \ dt' 



the latter differential coefficient will vanish between limits when k is 

 any number from to r inclusive, because it will always contain the 

 factor {tt'Y~'"^^ ; also when n and m are unequal we may suppose w to 



d'P„ 



be the greater, and since ft'' — j-^ is of in + ;• dimensions, it follows 



that if k> n + r, then k -1> in + 7-; and consequently the latter dif- 

 ferential coefficient will be identically zero. 



^*-i / d' P 

 The only instance in which the factor , .._, iff' , .'" j does not 



Aanish between limits is, therefore, where k lies between r + 1 and r-\-n 

 inclusive, but then the first factor is changed to ft'''P„; and since k — r 

 is now some immber from 1 to « inclusive, this factor vanishes between 

 limits (vid. Art. 5.), and therefore the part outside the sign of integration 

 vanishes in all cases, and we thus obtain , 



f,R.R.„ -(-1; j^-^^,-^.-^[tt -^jr)' 



