IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 23 



^ (ft - I) ! d^-ty, 



Z ( n _ k ) I $ _ q J °»-« rf^fe-*' * - 0, 1, .... n, 



are absolute co variants. Note that for k = n the expression reduces 



to L(u). This result is simply a translation into terms of differential 



expressions of the corresponding facts in the case of ordinary differential 



equations given by Wilczynski, Chapter II, § 2.5 And what follows 



is a mere extension to the case of partial differential expressions. 



The formulas, (17), expressing the as in terms of the a's may be 



given a form more advantageous for some purposes by introducing, in 



the coefficients of L(u), A(?/), further binomial coefficients. Let us 



ft! 

 put a pq = - — _ j.\] j.\ c pq> p + q = n — k, and, correspondingly, 



n! 



a pq = t 7 . , T , y vq - L(u) th us becomes, 



(n — k)lkl 



Tr , xV nl dn ~ ku 



while the formulas of transformation are 



ypQ ~^ fl lHHk-l-i)l Cp+i,q+k '' l - i 3*%*-*-*' 



p + q = n-k, (20) 



formulas in which everything except the subscripts of the c's is inde- 

 pendent of ft. Now let Lj(u) be an expression of the jth order, j ^ n, 



If we make the change of variable u=$-v, the coefficients of the 

 transformed expression will, by (20), be given by 



b v* - Z ZiM](h.-i-iw d p+i.i+k-i-i ^i^,k-i-i> p + q = 3-k- 



Z=0 i=0 



lH\(k-l-i)l a P+^+*-l-i dx i dy k-l-i> 



Now take any two numbers p, q such that p+ q — n — j. If we put 

 d vq = Cp + p iq+ g for all values of p, q such that p-h q^j, the expression 

 just written for 8 pq goes over into 



5 These covariants were first given by Cockle, Phil. Mag., 30 (1865) ; see 

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



