IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 35 



of yjr to be calculated in any way whatever in accordance only with the 

 rule above. These expressions, as we see, will be times polynomials 

 in k v « 2 and their derivatives. If, finally, we divide the whole by 0, 

 we get a rational function of the a's and their derivatives, and it is 

 this latter expression that we wish to prove an invariant of the differ- 

 ential equation. What we have to prove, then, may be stated in the 

 proposition : 



Proposition 17. The expression 

 1 i ^ (n - Q 1 a d*-W 



e£ Q £ Q (ji-k)\i\(k-i-i)\ ap+i ' 9+ *~^ a*V-*-« ' 



p + q = n — k, 



is an invariant of the differential equation L(u) = 0, where the deriva- 

 tives of are obtained from 



f = „«, £ = „» (34) 



dx dy 



by the rule above, and k±, k 2 are defined by (30). It is an invariant of 

 degree one. 



First, it is an invariant for a multiplication of L(u) by <f>. For k\, * 2 , 

 and therefore their derivatives also, are absolute invariants for this 

 transformation. So too, then, is any derivative of divided by 0; 

 while finally each of these latter expressions is multiplied by an a. 



Next we have to prove that our expression (33) is an invariant for 

 u = ty"ri. To this end let us turn back to the absolute covariants of 

 proposition 9, page 24. If we divide any one of these by (n — j) ! u, we 

 get a covariant of the first degree, which, by a change of notation, 

 we may write 



1 * *z? (n - /) I d k -hi 

 ~ > > *_*_;)! VK— ' 



p + q — n — k. 



u Q -3 (n - h) It I (* - / - i) 1 p+i ' 9+fc " Z_i d&dy*-+-i ' {66) 



Here we note the close analogy in form with (33). In fact, this co- 

 variant may be obtained from the formula (17) for a pq , reproduced on 

 page 34 above, by the substitution for the derivatives of yjr of the cor- 

 responding derivatives of u divided by u, just as (35) is obtained from 

 the same formula by the substitution of certain polynomials in k\, K2, 

 and their derivatives. 



