INTO ARITHMETICAL EXPRESSIONS FOR INFINITE SERIES. 189 



0.r = fix + y.v.<pax 

 (pax = fiax + yax. (pa'^x 



(pa"x = /3a" .r + ya"x . (puT^'x, 



whence, for a finite number of terms, 



(pX - yx .yaX^.^a^X (Pa"*' X = (ix + yX.(iaX+ ... + yX .yaX.. .ya''-'x . (ia'x. 



The first rule of which is a case of 



HX + vX.laX - 7A-.7aa;. ...ya'-xiy^a'^^'x ■\- va"*'x .'^a"*" X). 



In which >'X =^ yx .vaX = yx .yaX . ...ya"Xva''*'x, 

 and lax = ^a^K = ... = ^«" + 'x, 



whence the expression in question becomes 



fix — yx . 7a;i; . ...ya'"X.na"*'x, 



^vhich cannot, as might appear at first sight, give a different vahie for 

 every different vahie of txx: for, since two values of (px can only 

 differ by some solution of (px = yx .(pax, the preceding expression is 

 the same whatever value of (px be adopted. 



For an infinite number of terms of the preceding series, tve have 

 ixx — yx .yaX . ...ya^X . na'=' X. 



And the equation (px = yx.(pax, if „j- be one solution, can have no 

 others, except of the form vx . ^ax. But 



yX .yaX . ...ya'^X 



evidently satisfies (px = yX(pax; or if we take vx = yx . I'ux, we find 

 the preceding product to be vx -^ va"x. Consequently, the expression 

 for the series in question is 



vX ,ia" X 



IJ.X - —--. ixa^ X, or ixX vX. 



vvTX vu'^-X 



