V. TRANSFORMATION OF QUADRATIC FORMS. 573 



for which a rs = a sr , and 



K 2/i5 2/2? 2/w are another set of variables, and we put for each x r 

 the value x r + ly r we have 



a rs (x r -f- ^/ r )(^ s 



r = l * = 1 



r = n s=n r = n s=-n 



8) 



If now -R, the determinant of the form f vanishes, we have a relation 



r = n 



9) c^ + c 2 ^ 2 H ----- h c w w n EE 'fyfffa, . . . a? w ) = 



for W values of # 1? ... a; fl . 



Let us now put for the y's of equation 8) the values of c of 

 equation 9). We then have by 7) and 9) 



^ri . - O = 0, 



r=l 



so that 8) becomes 



10) + 



We thus find that in this case f is independent of I and of y l , . . . y n . 



x n 

 Accordingly if c n is not zero, let us put I = ? so that we obtain 



C n 



for all values of x lt . . . x n , 



C C C n l 



11 J / (jj, . . . X n ) = t(X^ X n , #2 X n , . . . X n i - / Q\ 



c ^n ^n 



Thus we obtain the theorem: every quadratic form in n variables 

 whose determinant vanishes, may be expressed as a quadratic form of 

 less than n other variables /, which are linear combinations of the original 



