450 Mr Dixon, On Partial Fractions. 



The method fails when three of the lines u = are concurrent, 

 and a simple case such as the fraction \jxy{x + y) shews that 

 the theorem is not generally true then. 



Suppose, as an example, that ?< 1 = 0, u 2 = 0, u 3 = 0, w 4 = all 

 meet in a point C, through which none of the other lines pass. 

 Then we may determine B 3i , so that 



U — B 3i u 5 u 6 u 7 ... vanishes at C, 



and then B 23 , B 2i , so that the curve 



U — B^u^Uq ... — B 23 u 4 u 5 u s ... — B 2i u 3 u 5 u s = 



has a node at G, and then B l2 , B 13 , B u , so that 



U - (B 3i + B 23 Ui + B 2i u 3 + B 12 u 3 u 4 + B 13 u 2 Ui + B u u 2 u 3 ) u 5 u 6 . . . = 



has a triple point at G. Then the partial fractions will be 



B 12 B 13 B u B., 3 B 0i B 3i 



— — + — + — + — + — H — + 



U-iU 2 U Y U 3 Ux'Ui U-iU 2 U 3 U-^UoUi l^Uo.U^i 



others found as before, with one or two of the expressions 

 u 5 , u 6 , u 7 , ... in each denominator. 



Similarly, if the first in of the lines are concurrent, we have, 

 instead of ^m(rn — 1) fractions with denominators 



u r u s (r, s = 1, 2, ... m), 



m — 1 with denominators, say, 



m — 2 with denominators 

 ??^ — 3 with denominators 



^1^2^'3^4 ? ^'l ^2 ^3 ^5 > • • ■ ^1 wo ^3 wj^ } 



and so on. The numerators are constants, and there are in all 

 -|-to (m — 1) of them. The form of the fractions in which one 

 factor of the denominator is taken from u 1} u 2 , ... u m and the other 

 from u m+1> u m+2 , ... u n , is not affected. 



The denominator may contain repeated factors. In such a case 

 some of the numerators will generally not be constants ; they 

 may be qualities in one of the variables only or in an arbitrary 

 linear combination of them. Suppose the variable x to be chosen, 

 and let the denominator contain u x in the m th power only, 

 u m+1 , u m+2 , ...u n being different and no three of the lines con- 

 current. Then the m— 1 coefficients in an (m — 2) ic , <f>(x), and 

 constants B, may be so chosen that 



77 A* ( \ -"m+i . X» m -f2 . ^ 



