IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 45 



for a partial differential expression of the obvious invariant -=- of an 



ordinary differential expression. For n = 2, 



„, . ^ da da 



§ 14. Reduction to Canonical Form of an Ordinary Differential 



Expression. 



We may obtain, in the case of an ordinary differential expression, 

 a system of rational invariants in terms of which all others may be 

 rationally expressed, by the same device as that employed, § 9, 

 for change of dependent variable. For let the change of variable, 

 £ = \(x), reduce the ordinary differential expression 



L(u) — a n u w + . . . + aou 

 to a canonical form with coefficients A . We are to have 



or 



x " = ~ n (» - 1) "tr x ' * (41) 



Hence any derivative of % is %' times a rational function of the a's 

 and their derivatives, and it follows that ^4 n _fc is (%') n— k times such 

 a function. For let L{u) go over under any transformation, £ = <f)(x), 

 into an expression with coefficients a. Then the formula 



d k u -^ , d l u 

 dx~k ~ ff x ' l dji ' 



fi being a polynomial, homogeneous of degree /, in the derivatives of 

 <f>, may be established by mathematical induction. Hence a~i is not only 

 linear in the a's, but homogeneous of degree I in the derivatives of £. 

 We have, then, that A n —k = (x') n— k Jn—k (a), Jn—k being a rational 

 function. It follows, just as in the similar case of § 9, that J n —k 

 is an invariant of weight n — k. 



Now let I be any invariant of weight w. Then 



\X ) *■ \fyij ®n > • • • <^n — 1) Q>n — 1 > • • • On — kj • • • &n — k> • • •) 



-J ( I d Az no I VAn-k \ 



— ■» I ^iji) ii , . . . U, U, . . . /in — k> ' • • 7>j i • • • 1 > 



