Mr Dixon, On Partial Fractions. 451 



contains the factor u x , provided of course that Wj is not inde- 

 pendent of y. Then 



U <f> ( x ) _ V -Bm+a 



is equal to a fraction in whose denominator only the (m— l) th 

 power of M x occurs. This fraction can be treated in like manner 

 and thus the original fraction is reduced to a series of fractions : 

 the denominators are 



iif (r = 2, 3, ... m), 



U\U m+s (r =1, 2, ... m; s = 1, 2, . . . n - m), 



Um+rUm+s (r, S = 1, 2, . . . 7b — 7Tb). 



The numerators in the second and third series are constant : in 

 the first series the numerators are quantics in x only, of degrees 

 lower by two than the respective denominators. 



If the lines v^ = 0, u. 2 = 0, ... are concurrent and the denomi- 

 nator contains the factors u 1} u 2 , ■■■ in the degrees m x , m 2 , ..., 

 then the partial fractions corresponding will have denominators 

 of the form 



where n x ^» m 1} n 2 ^> m 2 , . . . and ?? a + n. 2 + . . . <£ 2. 



If x does not vanish at the point of concurrence the nume- 

 rators may all be made functions of x only, each of degree lower 

 by 2 than the corresponding denominator. Some of the coefficients 

 in these numerators may be arbitrarily chosen : a general rule 

 for finding the others would be to express the numerator U and 

 the other factors of the denominator as homogeneous functions 

 of x, u x , u 2 , and then divide U by the product of these other 

 factors in the denominator, in descending powers of x : each power 

 of x in the quotient could then be treated by itself. 



Thus a series of partial fractions would arise from each point 

 of concurrence of two or more of the denominator lines : the 

 sum of all the partial fractions is the original fraction. The 

 case with which we started is that in which each series contains 

 one fraction only. 



If »!, u 2 , ... u n are linear expressions in p variables and U of 



the degree n—p, then U/ujU 2 ... u n can be similarly resolved into 



partial fractions. The denominator of each will contain p factors 



n ! 



and the numerators will be constant : their number is —r—. — , , 



p ! (n — p) ! 



the same as that of the coefficients in U. The exceptional cases 



are more complicated and seemingly of no great interest. 



30—2 



