IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 7 



Lemma. If N(v) be a linear differential expression, and T an ex- 

 pression bilinear in u, v, and their derivatives, and if 



at/ \ dT 

 uN(v) =dx~' 



then N(v) = 0. 



Since Lagrange's Identity may be written 



d(-S) 



uM(v) — vL{u) 



dx 



we infer that L(u) is the adjoint of M{v) : the relation between an ex- 

 pression and its adjoint is reciprocal. 



A multiplier of L(u) is defined to be a function, v(x), such that 

 vL(u) is a derivative of a differential expression of the (n — l)st order, 



i( \ dP 



The condition that v should be a multiplier of L(u) is that v should 

 satisfy the differential equation M(v) = 0. The sufficiency of the con- 

 dition is obvious from Lagrange's Identity; its necessity follows from 

 an application of the lemma to 



.,, . d(P - S) 



For conditions that L(u) should be self-adjoint, when n is even, the 

 negative of its adjoint, when n is odd, that is, L(u) = (— \) n M{u), see 

 below, page 15. The problem of making L{u) equal to (— l) n times its 

 adjoint by multiplying it by a suitable function of x will occupy us 

 later. 



§ 2. Partial Differential Expressions of the Second Order. 



We take up next the theory of the adjoint for partial differential ex- 

 pressions, and here a somewhat different order of presentation will be 

 found advantageous. We consider first expressions of the second order. 



Let L(u) be such an expression, 



m ^2 Wl -\ 



L(u) = 2 a a 5-^— + 2 a i 5— + au > ( 4 ) 



ij^ t dxidxj pi dxi 



