452 Mr Dixon, On Partial Fractions. 



The process is of use in discussing the reduction of a ternary 

 n ie V to the form of a symmetrical determinant with linear 

 constituents. 



Project to infinity a line meeting the curve V = in n separate 

 points, no three of the tangents at which meet at a point. This is 

 always possible with a proper n ia curve, for if not, let A 1} A 2 , ...A n 

 be the points where it meets an arbitrary line, and suppose the 

 tangents at A 1} A 2 , A 3 to be concurrent in all positions. Let 

 the line turn about A x and come back in such a way that A 2 

 comes to the position A±. Then the tangents at A 1} A± and 

 another of the n points must be concurrent. Similarly, if any 

 two suffixes r, s are taken there is a third, t, such that the 

 tangents at A r , A s , A t meet in a point, and this holds in all 

 positions of the straight line. Let A 2 move up to A 1 so that the 

 line becomes a tangent. Then the tangent at A s must pass 

 through A ly since this is the point of intersection of the tangents 

 at A 1} A 2 , and the line is therefore a bitangent. Every tangent 

 is therefore a bitangent and meetsthe consecutive in two points : 

 it must therefore coincide with the consecutive and the curve 

 must break up into straight lines. 



We may then write V=u 1 u 2 ...u n — U, where Wj = 0, u 2 = 0... 

 are the asymptotes of V= 0, no three of which meet in a point, 

 and U is of degree n — 2 at most. In the exceptional case U 

 vanishes identically, and V is equal to a determinant having 

 u 1} u 2 ...u n in the diagonal and in other places. 



When U does not vanish identically, assume 



u x u 2 ... u n — \U = 



Cfe 



a symmetrical determinant of order n in which a 12 , a 13 ... are in- 

 dependent of x, y : these quantities are functions of X, and when 

 X = the conditions are satisfied by taking them all to vanish. 

 Suppose X to be small, and a^, a 13 ... also small. Then we have 

 for a first approximation, neglecting powers and products of 

 a 12 ,a 13 ... above the second, 



X U = u ± u 2 ... u n .X a rs 2 j u r u s , 



or with our former notation 



a r8 = A^i. rs . 



The quantities a rs are thus all found to a first approximation. 

 The approximation can be carried to any degree, and as the 

 equations to be solved are algebraic the value of a rs can be found 

 as a series of ascending powers of ^/\ for values of X within 



