Mr Bromwich, The Classification of Conies and Quadrics. 363 



It should be observed that any number of d's may be equal 

 without affecting the form of G; for all the invariant-factors of A x 

 {Elementartheiler) must be linear 1 , and then the form of G will 

 still be the same though we are left with some arbitrary quantities 

 in the y's (two simple illustrations of the geometrical meaning of 

 this arbitrariness are given by considering a surface of revolution 

 and a sphere in space of three dimensions). 



Expanding G we see that the negative powers of \ give 



or SU = - %,"/«„ GL, = - Sy,.', (r = 1, 2, . .'., h). 



r r 



We shall next require the coefficient of A, and the term indepen- 

 dent of A, in the expansion of F in powers of (1/X); these can 

 be found by a method given in my note quoted above (p. 358) 

 which is analogous to the Weierstrass-Darboux process. Since 

 the invariant-factors of A x which correspond to the infinite root 

 are here linear, this will be found to give no term in X and the 

 term independent of X will be 



%z r -, (r = 1, 2, .. ., n — h — 1). 



)• 



From this it appears that 



F= ~%z.f + negative powers of X. 



r 



and there are no positive powers of X in F. 

 Thus /3 = and hence a?/3 + 2?2. 



Now returning to the values of A, B found in Section 1, we 

 shall find, on putting p x =l, 



^ = % W . + 2 fl -8 I h 



S-Sy, , + 2».' + 2&-8, U = l,2,.-.,n-A-l/' 



r « 



1 Aj is the determinant of (\^i - .B) when the .c's are subject to the condition 

 "2,p r x r = 0; or, more precisely, if x n be eliminated from (\A - B) by this condition, 



the determinant of tbe resulting quadratic form of (n - 1) variables is Aj multiplied 

 by a power of p n . When this elimination is made in B the resulting form is 

 definite (in the variables af lf ... , x n _ 1 ) ; thus by a theorem due to Weierstrass (Berl. 

 Berichte, 1858 = Ges. IVerke, 1, p. 233) it is clear that the invariant-factors of Aj are 

 all real and linear. Since these are real, all our transformations will be real ; and 

 further A x (0)^0, so that all the 0's are different from zero. Since the invariant- 

 factors are all linear, it follows from the investigation of Weierstrass (Berl. Berichte, 

 1868 = Ges. Werke, 2, p. 19) and others, that the reduced form of G is the same as 

 if all the factors of A 2 were distinct. If h < (n - 1) there will be infinite roots of 

 Aj = ; the corresponding invariant factors are found by taking those to base p. of 

 the determinant obtained by writing c rs = a rs - fib rs in Aj ; these are linear by the 

 same argument. 



