390 Mr MURPHY'S THIRD MEMOIR ON THE 



PAGE 



Section III. Application of the preceding principles to the Phaenomena of Developed 



Electricity/ 386 



Note (A), No. 2. On the general separation of the positive powers of the variable from 



the negative 402 



Note (B), No. 1 . On the apparently improper forms of (p (x) 404 



No. 2. Method of valuing the results of operative functions 406 



SECOND MEMOIR. Vol. V. Page 113, &c. 

 Introduction 113 



Section IV. Inverse Method Jbr Defitiite Integrals which vanish, and theory of Reci- 

 procal Functions. 



Arts. 1,2. X being restricted to the natural numbers 0, 1, 2, (»— 1) to &nd fit) so 



tha.tf,f(t).f = 116 



Art 3. P„ denoting the function y(<) above-found, when m and n are unequal /,P„P„=0, 



and when equal /,P,P„ = 117 



Art. 4. To find a rational function _/(<) which may satisfy the equationy^y(<)'''=0, x being 



any number of the series p, ^+ 1,. . .p-{-n—l 118 



Art. 5. The general form of fit), when x is from to n— 1 inclusive, is 



d" (ft'" V\ 

 At) = • ^^^„ ^ , where t' = \ - t 118 



Art. 6. In this case the equation^" (/) = 0, has n real roots lying between and 1 1 19 



Arts. 7, 8. To find a rational function of h. 1. /, such that /,y{h. 1. (<)}.<' = 0, when 



X is from to n— 1 inclusive 120 



Art. 9. Denoting this function by Z.„, the function which it generates is the value of 



u in the equation u {l — hh.\. u) = t 122 



du 

 Art. 10. If Q„ be the coefficient of *" in -r- , u being found from the equation 



m(1 — A £/) = <, where J7 is a function of u vanishing when u = \, and T the same 



function of t, then -^l = "("- V""^"~""^'^ 122 



Art. 11. If t/ be a rational and entire function of u vanishing when m=1, and Q, be 

 the term independent of u in the product i/" I 1 I , then shaliy]Q„<'= 0, when 



X is from to «— 1 inclusive 12S 



Art. 12. To find (p, q)„ a rational and entire function of P" of n dimensions, which 



multiplied by a rational and entire function of f of less than n dimensions, the integral 



of the product may vanish from t — Otot=l 125 



Art. 13. Reciprocal Functions; such are ip,q)„, {<l>p\'; value of the integral of the 



product when n=n' 126 



Art. 14. To find a function A„ reciprocal to the function L„ found in Art. 8 128 



