574 



NOTES. 



variables. Such forms are called singular, as opposed to ordinary forms, 

 whose determinant does not vanish. As an example the form 



^2 * ^3 ' 



111 

 1 1 

 0-1 



whose determinant 



vanishes, may be written 



If f is an ordinary form and a rr is not zero, we may write 



12) f=a rr x r *+2x r p + q, 



where p is a linear form containing all the other variables except x r 

 and q is a quadratic form in the same variables. Completing the square 

 we may then write 



13) a rr f = (a rr x r -f p) 2 + a rr g p 2 . 



If on the other hand every coefficient of a square a rr is zero, we may write 



14) f= 2a rs x r x s -f 2x r p -f 2# g (? + r, 



where J9, # are linear, r a quadratic form not containing either x r or # 

 we may then write 





2 /"= 4 (a^ov 



-f 2 a rA . 



In the former case we have exhibited f as the sum of a square and a 

 form in w 1 variables, in the latter as two terms in squares and a 

 form in n 2 variables. In either case the remaining form may be 

 treated in a similar manner, and so on, so that the form is eventually 

 reduced to a sum of terms in squares. If the coefficients A of all the 

 squares in 



16) f 



when the y's are linear forms in x v . . . #, have the same sign, the 

 form is said to be definite, for it can not change sign however the values 

 of the variables be altered. If the coefficients A are not all of the same 

 sign the form is indefinite. 



If we transform the linear forms l) by means of the linear sub- 

 stitution 



