572 



NOTES. 



NOTE V. 



TRANSFORMATION OF QUADRATIC FORMS. 



The last two notes have dealt with quadratic forms, and in Note IV 

 we have by a linear transformation of the variables 19) transformed 

 the form F into a form 20) in which no product terms appear, and we 

 find that the coefficients of the squares are the roots of the determinant 6). 

 In this note we shall consider similar transformations of forms of any 

 number of variables, and shall incidentally obtain a proof of the reality 

 of the roots of Lagrange's determinant, 65), page 159, for the case of 

 no dissipation. 



We shall require a number of elementary properties of both linear* 

 and quadratic forms, which we shall now set forth. Suppose we have n 

 linear forms 



1) 



a 2n cc n , 



and let us call jR the determinant 



Clio. 



E = 



"11 



If we multiply the & th column of E by ##, and then add to this column 

 the first, second, etc., multiplied respectively by # 1T # 2 , 



Ol,*-f li Oi ? 



we obtain 



3) 



#1,* 1, 



a nn 



If now the determinant R is zero, the determinant on the right vanishes T 

 expanding which we obtain 



4) c^ -h c%u 2 + h c n u n = 0, 



where the c's are the minors of the elements of the fc th column of E. 

 Thus if the determinant of the forms vanishes, the forms are not independent, 

 but satisfy identically the linear relation 4). 

 Consider now the quadratic form 



r = n s=n 



