IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 27 



differential equation is b n and essentially the only invariants of the 

 second degree are b n 2 and 



, , n(n—l) fdb n , , db n -{\ n — 1 , „ 

 nb n b n -2 + —^- [-^ 6,-1 - b n -j- j ^W* 



, n(n — 1) /rfa n 6?a n _i\ n — 1 



= nanOn-2 + 2~ - ^ "- 1 " a » ^T J - ~2~ a 



which is na n times the invariant called I n —2, (23) below. 



Essentially the only invariants of the first degree of a partial differ- 

 ential equation of the second order in two independent variables are 

 &n, b i2 , b 2 2, and of the second degree, besides powers and products of 

 the bi/s, those of the form 



dbij db kt dbij db H 



-Z— UJcl — Oij — — , — k i — b{j —— , 



ox ox oy oy 



invariants which involve the b's with two subscripts only. 



TIL Reduction to Canonical Form. 



§ 9. Ordinary Differential Expressions. 



Another method of treating the problem of invariants, a method 

 that applies to the case of an ordinary differential expression, is to 

 reduce that expression by a suitable change of dependent variable, 

 to what we shall call its canonical form, namely, a form in which the 

 coefficient of the (n — l)st derivative is zero. 



The corresponding investigation for the case of ordinary differential 

 equations will be found in Wilczynski, Chapter II, § 2.6 The treat- 

 ment of the two cases is, to a large extent, identical ; so that what fol- 

 lows is given not so much for its own sake as because a number of 

 the results admit of extension to partial differential expressions. 



Suppose then we have an ordinary differential expression 



L(u) = a n u^ + ....+ crow, 



accents denoting differentiation. Let it be reduced, by the change of 

 variable u= 6-v, to canonical form 



A(t?) = A n r,M + A n - 2 -n {n - 2) + . . • . + Aqv 



6 We note, to avoid confusion, that Wilczynski calls our canonical form 

 semi-canonical. The method is due to Cockle, Philosophical Magazine, 39 

 (1870); see Bouton's paper in the Amer. Jour, of Math., 21 (1899). 



