IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 51 



riant for a change of dependent variable; that is, that if L(u) go over, 

 under u = ty-y, into A(t;), then A(?7) may be made equal to (— l) re 

 times its adjoint by multiplying it by <f>i}r. That property, then, per- 

 sists under all these transformations. In parallelism with this fact, 

 the conditions we shall obtain for its existence are the vanishing of 

 expressions invariant under all these transformations. 



Taking first the case of ordinary differential expressions, let us begin 

 with those of the second order. The condition that 



L(u) = anu" + a\u' + au 



should be self-adjoint is, by (10), page 10, a\ = an'. The condition 

 that 4>-L{u) should be self-adjoint is, therefore, 



<£ai = -T- (<£aii), 



or a\\4> + (an' — «i) <£ = 0. 



It is always possible, then, to make an ordinary differential expression 

 of the second order self-adjoint by multiplying it by a function of x; 

 the latter function has merely to be a solution of the differential equation 

 last written. 



We note that, since -=- (<£an) = 4>a\, 4> • L(u) may be written in the 

 form 



4>.L{u)=j-{Ru!) + Gu, 



where K = <f>au, G = <f>a, and </> is determined as above. A differ- 

 ential equation, then, 



u" + pu' + qu — 0, 

 may be thrown into the form 



^ (7vV) + G(u) = 0, 



where K = $, G = <f>q = Kq, and ^> is a solution of <£' = p<f>, or, 

 say, (f> = e Svdx . This is Sturm's Normal Form for such an equation. 



For ordinary differential equations of the nth order, the solution 

 of our problem will be found in Wilczynski, page 46. The conditions 

 there obtained consist in the vanishing of the so-called linear inva- 

 riants of odd weight, that is, in Wilczynski's notation, of 03, ©5, etc. 



