V. TRANSFORMATION OF QUADRATIC FORMS. 



577 



(If the sum ^ J 



r=l *=1 



must have 



a rs c r y is is zero for all values of y 1? . . . y n -\, we 



for 5=1, 2, , ... n 1, but since f(c^ . . . c n ) is zero we must have 



r=n 



also ^ a rn c r , and these n equations require the determinant of the form 



r=l 



to vanish, and the form is singular.) 



As a result of this theorem we see that ifji form is to be definite, 

 no coefficient a rr of a square x r 2 must be absent, and all must have the 

 same sign*. For if a rr = 0, putting all the variables equal to zero 

 except x r would make the form vanish, and if a rr is not zero, the same 

 assumption would make the form have the sign of a rr . Consequently all 

 these coefficients must be of the same sign. 



Let us now consider two ordinary quadratic forms of the same 

 variables, with real coefficients 1 ) 



r=l s = l r = l s=l 



from which with an arbitrary multiplier A we construct the form 

 28) Ay -f i/;. 



As we give A an infinite set of real values, we obtain an infinite sheaf 

 of forms. Let us examine whether they are definite or not. 

 The determinant of the form Ago -f i/; 



28) 



-f- 



-f 



-f 



-f 



c nn 



= f 



is identical with Lagrange's determinant, page 159, when the %'& are zero. 

 (We here have written I for the A 2 on p. 159.) We shall now prove 

 that if the equation f(k) = has a complex root, all the forms of the 

 sheaf Ago -f- i/; are indefinite. 



Let A = a -f if be a complex root of the determinantal equation 

 /"(A) = 0. Then since the form (a 4- if) y -f ^ is singular, it may be 

 represented as a sum of less than n squares, and since it is complex, 

 these may be squares of complex variables, so that we have 



29) (a 



1) Kronecker, Uber Schaaren quadratischer Formen. Monatsber. der Konigl. 

 PreuB. Akad. d. Wiss. zu Berlin, 1868. pp. 339 346. Werke, Bd. I, p. 165. 



WEBSTER, Dynamics. 



37 



