V. TRANSFORMATION OF QUADRATIC FORMS. 579 



A linear divisor such as A K r of the determinant of the form 

 A g/ -|- ty r is also a divisor of the determinant of A g> -f- i/;, for on writing 

 out the determinant of the form 35) in terms of y^ . . . y n ^\ t s y .^ we find 



36) a^ to . . . ,,_,(* ?> + *) = (x - i.) tf* . . . r ._,(V + *0, 



so that the vanishing of the determinant of order n 1 on the right 

 makes the determinant on the left vanish. But this equal to the deter- 

 minant of A (p -f- ty in the variables x , . . . x n multiplied by a constant. 

 We may now treat the form hep' -\- i^ f in the same manner, and so 

 on, so that finally we obtain 



37) lg> + y = (I - Aj^ 2 + (I - A 2 > 2 2 + - - - + (A- A w )^ 2 , 



where A 15 . . . K n are the roots of the determinantal equation f(k) = 0. 

 Since this is true for all values of A we have 



which is the simultaneous transformation of two quadratic forms required 

 in the treatment of principal coordinates. It is obvious according to 

 this method that it makes no difference whether the determinant has 

 equal roots or not. 



