203 



Prof. H. T. S. Smith on Quadratic Forms [April 21, 



whatever be the real transformation by which the reduction is effected. 

 For the index of inertia we may take the number of the positive squares ; 

 it is equal to the number of continuations of sign in a series of ascending 

 principal minors of the matrix of the quadratic form, the series com- 

 mencing with unity, i. e. with a minor of order 0, and each minor being 

 so taken as to contain that which precedes it in the series (see Professor 

 Sylvester " On Formulae connected with Sturm's Theorem," Phil. Trans, 

 vol. cxliii. p. 481). The distribution of these forms into Orders depends 

 on the following principle : — 



Two forms belong to the same order when the primary and secondary 

 divisors of their corresponding concomitants are identical." 



Since, as has been just pointed out, there are, beside the fundamental 

 concomitants, an infinite number of other concomitants, it is important to 

 know whether, in order to obtain the distribution into orders, it is, or is 

 not, necessary to consider those other concomitants. With regard to the 

 primary divisors, it can be shown that it is unnecessary to consider any 

 concomitants other than the fundamental ones ; ^. e. it can be shown that 

 the equality of the primary divisors of the corresponding fundamental 

 concomitants of two quadratic forms, implies the equality of the primary 

 divisors of all their corresponding concomitants. And it is probable (but 

 it seems difficult to prove) that the same thing is true for the secondary 

 divisors also. 



Confining our attention (in the next place) to the forms contained in 

 any given order, we proceed to indicate the principle from which the sub- 

 division of that order into genera is deducible. 



If Fi be any quadratic form containing r indeterminates, and be its 

 concomitant of the second species, we have the identical equation 



F,(a.„ ...... ^,) X FXy,, y„ . . . 



(B) 



in which the symbol F^ (^^^* ' ' ' indicates that the deter- 



u 2» • • •> r\ taken for the indeterminates of 



F2, the order in which they are taken being the same as the order 

 in which the determinants of any two horizontal rows of the matrix 



of F^ are taken in forming the matrix of Fg. Let Qi—^f^ for every 



value of i from 1 to w— 1 ; it will be found that, if we form the concomi- 

 tant of the second species of 0, its primary divisor is the quotient 



^^-r— which, as has been shown elsewhere (see Phil. Trans. 

 Vi Vi-i 



vol. cli. p. 317) is always an integral number. Let li be any uneven 



