Mr Dixon, On Partial Fractions. 453 



a certain range, not infinitesimal. Beyond this range the values 

 can be analytically continued if necessary, and the solution thus 

 found when X = l. Hence any ternary quantic can be expressed 

 as a symmetrical determinant with linear constituents : the 

 problem is not poristic. (Compare Proceedings, Vol. xi. p. 351.) 

 Since when A, is small each of the quantities a rs has an 

 ambiguous sign, the number of solutions indicated is 2* n(w-1) ; 

 since however in any symmetrical determinant the sign of any 

 constituent in the first row, after the first, may be changed at 

 will by changing the sign of its row and the corresponding 

 column, only one in 2' 1 ~ 1 of these solutions is to be counted: 

 there are therefore 2^ n ~ 1 ^ n ~ 2) distinct solutions at least. That 

 there may be more is indicated by the fact that 2u 1 u 2 u 3 can be 

 expressed in either of the forms 



, 2th 

 u 2 



u. 



u 3 u 2 

 u 3 Wi 



U n Mi 



That all the 2^ n-1 X w-2 ) solutions given by the process are 

 really distinct follows from a consideration of the terms of the 

 third degree in a 12 , a 13 .... These will be altered by any change 

 in the signs of o\ 2 , a ls ... other than those we have allowed for. 

 There are, for instance, two terms a 12 a 13 a 23 W4^ 5 w 6 ... in the ex- 

 panded determinant, and if these are to be unaffected then two 

 or none of a 12 , a 13 , a 23 must be changed. Similarly for a rs , a st , a rt . 

 Let the suffixes > 1 be divided into two classes, r being of the 

 first class when a lr is changed and of the second when a lr is un- 

 changed. Then a rs is changed if r, s are of different classes, and 

 unchanged if they are of the same. Thus the rows and columns 

 with suffixes of the first class are changed and the others un- 

 changed, which was to be proved. 



