IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 43 



On the other hand, 



du du ^ d$k d£i du du 

 dxi dxj ~* dxi dxj d$k d${ 



It appears thus that the coefficient of 



dhi 



in 



d 2 u 



d$kd£l dxidXj 



is the same 



as the coefficient of — — - in . Now in calculating the a's 



d$ k d€i dxi dxj 



with two subscripts, a#, we are not concerned with the first deriv- 

 atives of u with respect to the x's or the £'s; so that the a i? 's are 

 expressed in terms of the ay's, in the case of L(u) under a change of 

 independent variables, by the same formulas as for the expression 



2 a a ^ — t: — under the same change of variables, that is, as for the 

 11 dXi dXj * 



quadratic algebraic form J> aijZiZj under the linear transformation 



= 2 



dxi 



*k' 



a linear transformation whose determinant, as we note, is J. 

 Now the discriminant 



A = 



a n 



aim 



a m \ 



a 



mm 



is a relative invariant of weight two of the algebraic form. A is 

 therefore also a relative invariant of weight two of the differential 

 expression, L{u). We note that A is also an invariant, for change 

 of dependent variable, of the differential equation. 



Again, if v\, . . . v m , w\, . . . w m be two sets of variables contra- 



gredient to the — 's, then 

 ox 



an 



aim vi 



a m i 



W\ 



a-mm Vm 

 Wm 



= 2 AijViWj, 



(37) 



i,i 



A a being the cofactor in A of a ij} is invariant of weight two of the 

 algebraic form, and therefore of L(u) also. Now the differentials of 



