382 Mb MURPHY'S THIRD MEMOIR ON THE 



Put F{k) = Fo + F,A + V-it + kc. ad infinitum, 

 as found in Art. 28. Sect. vii. 



Hence F{ - kr') = K- V.kr" + V.,kr' - &c. 



theretore j^ ^^^-— ^ - 2 ^^» a ^^*+ 2.4" '^'^ 2.4.6-'^'^''^ 



the limits of t being and 1 ; 



TT ^^ v'l-T^ »2' 2.4 2.4.6 



Ex. 6. j,<l>{t).f{a-t)=f{a-h). 



Denote by Pi,„ the reciprocal function P„ when ^ is the variable, 

 by Pi_ri when 6 is the variable. 



Let/(«-0 = J,P,,, + A,Pt,, + A,P,,, + A^Pt,^ + &c. 

 and (j>{t) = CoPf.o + c,Pu + CaP,,^ + c^.Pt.z + &c. 



.-. f{a-h) = ^oCo + g . AxCi + g . ^sC^ + \ .A^Ci + &c. 

 but changing t into J in the expansion of J'{a — t) we get 



f{a-h) = AoP,,o + ^,Pm + A,P,,, + A,P,, + &c. 

 which values are identical when Cc = Pi.,o, c, = SPs,,, c^ = SPh.n, &c. 

 therefore (j){t) = PmP.o + SP^P., + 5P„,P,. + 7 P.^P.s + &c. 



45. Ow ^A^ appendage necessary to complete the Solution of' a 

 Definite-integral Equation. 



In the examples in which f{a, t) = cosa^ given in the last article, 

 the function F{a) is adapted to general differentiation relative to a, 

 under the definite integral ; but besides the prime value thus obtained, 

 there must be an appendage to represent the same operation on zero. 



