PROCEEDINGS 



OF THE 



damkiirg* f^ttosopjjkal Stotfelg. 



On Partial Fractions. By A. C. Dixon. 



[Received 28 March 1904.] 



Let u r — a r x + b r y + c r (r = l, 2, ... n) and let U be a rational 

 integral algebraic expression in x, y of degree n — 2; then, if no 

 three of the lines u^ = 0, u 2 = 0, ... meet in a point, the fraction 

 JJJu-^Uz ... u n can be expressed as the sum of partial fractions of 

 the form A rs /u r u s with constant numerators. 



For let the expression ~ZA rs /u r u s be brought to a common de- 

 nominator ; the numerator is an expression of the degree n—2 

 and contains \n(n—V) arbitrary coefficients, such as A rs , that is, 

 as many as there are coefficients in U. Also none of these arbitrary 

 coefficients are illusory since each may be uniquely determined. 

 To find A rs , put u r = = u s in the proposed identity 



V = u 1 u 2 ... u n 'SiA rs /u r Ug. 



The right-hand side is reduced to one term, containing A rs , and 

 this does not disappear, since no three of the lines u T = 0, u 2 =0, ... 

 are concurrent. Thus A rs is found. 



The identity proposed is thus established, but it may be verified 

 as follows. If the numerators A have the values found for them, 

 the expression 



U ™~ wy w> • • • tA/fi & xl fS i ***¥ ^S 



vanishes at n — 1 points on the line u x = 0, namely, those in which 

 this line meets u 2 = 0, u 3 = 0, ... u n = 0. Since the degree of the 

 expression is only n — 2, it must contain m x as a factor, and 

 similarly u 2 , u 3 , ... u n . It can therefore only vanish identically. 



VOL. XII. PT. vi. 30 



