V. TRANSFORMATION OF QUADRATIC FORMS. 575 



we obtain linear forms in the new variables y^ . . . y n , so that if we write 



2/1 + C 12 2/2 



18) 



= C 21 2/1 



H 



we find by carrying out. the transformation, 

 19) c rs -- 



But this is, according to the rule for multiplication of two determinants 

 the condition that the determinant of the forms u in y, 



20) 



is the product of the determinant of the forms in x by the determinant 

 of the substitution 17). 



The determinant in which the element in the r th row and s th column 



du r 



is a derivative * is called the Jacobian of the functions u< , . . . u n with 



dx s 



respect to the variables # 19 . . . # n , and is often denoted by 



?(,....*.) 



In this notation 20) becomes 



/ ~WT^i ^~\ = 



fti. 



If the functions M lt . . . tt n in a Jacobian are the partial derivatives 



o / 



of the same function /", w r = -jr^-* so that the element in the r th row 



gsr ^^r 



and s th column is -~ 5 the determinant is called the Hessian of the 

 dx r dx s 



function. Thus the determinant H of 2) is the Hessian of /", and will 

 be denoted by H x f. 



If now we transform the quadratic form 5) by the substitution 17), 

 so that 



22) 



