IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 15 



The first equation of (15) shows that a differential expression of 

 odd order cannot be self-adjoint, nor one of even order equal to the 

 negative of its adjoint. Let us call a differential expression that is 

 the negative of its adjoint, L(u) = — M(u), anti-self-adjoint. Then 

 we are led to inquire under what conditions a differential expression 

 will be self-adjoint or anti-self-adjoint, L(u) = (— l) n M(u). Such 

 conditions may be readily deduced from the symmetrical formulas (16). 

 For let 



t>PQ = (— !) n a PQ> V + q = n — I, 



for p = 0, 1, . . . (n — F), and for all values of I < k, k being a given 

 even integer. Then, on substituting these values in the left member 

 of (16), all the terms but the last on each side cancel, and we have 

 left 



Ko. — (— l) n a PQ> p + q = n — k, 



p = 0, 1, . . . (n — k). Hence, by mathematical induction, we ob- 

 tain the conditions (which are, of course, necessary) : 



Proposition 4. Necessary and sufficient conditions that a differen- 

 tial expression should be self-adjoint or anti-self-adjoint, as the case 

 may be, L(u) = (— l) n M(u), are that the coefficients of the(w — k)th 

 derivatives in L(u), should be ( — l) n times the corresponding co- 

 efficients of M(u) for all odd values of k. 



This proposition has already, in effect, been deduced for expressions 

 of the second order; cf. (10), obtained from the second equation of 

 (8) by putting h = a;. 



Lagrange's Identity. We may deduce for any differential expression 

 a formula similar to what we have called Lagrange's Identity, or 

 rather a great number of such formulas, by the following process: 



dhi 

 dxPdy4 

 for the coefficient. We have, to start with, 



Take any term of vL(u), va ^ v ^ , where we now write a simply 



dhi d ( b^-^u \ diva) d^hi 



va ~ — - — = — [ va 



dxPdyi dx \ dxP~ 1 dyi J dx dx^dyi 



d' 2 b n . _ d 2 b u , d 2 b 22 dbi db 2 . _, ,, , ,. , ,, , 



-j~ + 2 -j-~ + -~ — — — $ 2 + 2b = the same function of the a's, 



an equation which differs from the last equation of (9), written for the case 

 of two independent variables, by the presence of the terms in the second 

 derivatives; terms that cancel each other, indeed, on the two sides of the 

 equation just written. The remaining equations, n = 2, k = 1, given by (16) 

 agree with those of (9). 



