IRWIN. INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 9 



multiplier of L(u) is that it should satisfy the differential equation 

 M (v) = 0. 



The condition is also sufficient. For let v be any solution of 

 M(v) = 0. Then choose, for instance, the Pi/s for which i > ;' at 

 pleasure ; then the rest of the p^-'s and the p/s may be determined to 

 satisfy equations (6a) and (Qb). Equation (6c) will thereby be satis- 

 fied also, and we shall have 



vL(u) 



For if (6a) and (66) are satisfied, 



^ d 2 (aij-v) ^ d(a,iv) 





dpi 



t>? 



dxidxj 



dxi ^ dXi 



Now since M(v) = 0, the left side is equal to — av; that is, equation 

 (6c) is satisfied too, as asserted. These considerations show us that the 

 quantities P; on the right side of (5) are not uniquely determined by v 

 being given. We may state the result just obtained by saying : 



Proposition 1. A necessary and sufficient condition that v should be 

 a multiplier of L(u) is that v should satisfy the differential equation 

 Miv) = 0. 



If we write M(v) in expanded form, 



M{v) = ^b ij 



t.j 



d 2 v 

 dXidXj 



+ 2 bi ^ 



dxi 



then the b's, the coefficients of the adjoint, will be given by the 

 formulas 



°ij — a ij y 



° l "Z dx.j a% ' 



1,7 



dxidxi 



dXi 



+ a. 



(8) 



These equations may also be written in symmetrical form, 



y^i h = 



dxj 



dbi 



dxi 



2§ + ^ 



26 = 2,--2" 



*r< dXi 



(9) 



