IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 53 



VIZ 



dLi dLj _ d(\j — Ki) d(Xj — kj) _ fdM _ dAA /cK d*y \ 

 dxj dxi ' dXj dxi \dxj dxi J \dx 3 - dxi J ' 



that is, is equal, page 32, to the difference of an absolute invariant 

 and the adjoint invariant. 



ri J • rl T • 



The expressions ■ — are not, on the other hand, except in 



dxj dxi 



the case of two independent variables, invariants for a change of inde- 

 pendent variables. They are, however, the coefficients of what, to 

 extend somewhat the definition of that term, we may call a covariant, 



., ^ ( ■ — - — — - ) dxfixj, where the dx's and &x's are two inde- 

 fi \dxj dxi J 



pendent systems of differentials. Reserving for the moment, until we 

 have discussed partial differential expressions of the nth order, the 

 proof that the expression above is a covariant, we may state the solu- 

 tion of our problem, for the case in hand, as follows : 



Proposition 24. A necessary and sufficient condition for the possi- 

 bility of making a differential expression of the second order self- 

 adjoint by multiplying it by a function of the independent variables is, 

 if the invariant A does not vanish, the identical vanishing of the 

 expression 



2 Gi - W dxM> (43) 



the L's being defined by (42). The coefficients of this express'on, 

 — -, are absolute invariants, for change of dependent variable, 



of the differential equation, and the expression itself is absolutely in- 

 variant for change of independent variables. 



Let us look now for a moment at the case of partial differential ex- 

 pressions of the nth order. We take, as usual, for illustration, two inde- 

 pendent variables. In order, first, that the coefficients of the (n — l)st 

 derivatives in </> • L(u) should be (— l) n times the corresponding coeffi- 

 cients of its adjoint, we must, by (15), page 14, have 



p + q = n — 1, p = 0, 1, . . . (n - 1). (44) 



If these equations are to be solvable algebraically for 2_Z, — , 



dx dy 



it is necessary that all three-rowed determinants of the matrix 



