IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 47 



It remains to prove that J n -k,i is a rational invariant of weight 

 w — k — I. Since 



T 1 d[( X ') n - k - l Jn-k,l] 



Jnr-k,l+l ~ ^n-fc-J-l " d£ 



t i 2 (w — fc — /) a n -i 



— J n—k,l 7 TT - „ J n— k,h 



n{n — 1) a n 



the case in hand comes under the proposition : 

 Proposition 22. If I be an invariant of weight w, then 





is an invariant of weight w — 1. 



This proposition may be proved as follows. The expression in ques- 

 tion is equal to 



1 fna n I' — wan'I w ( , 2 \ T ~| 



a n \_ n n\ w—1 J _\ 



2 



Here a n ' a n —i=Jni, and is shown, by direct calculation, with 



w—1 



the help of the formulas 



a n = (4>') n a n , 



«n-l ={4>Y- 2 ( ^ {n ~ l) <t>"On + fan-l^, 



to be an invariant of weight w — 1. On the other hand, since I n /a n w is 

 an absolute invariant, its derivative is an invariant of weight — 1, 

 that is, na n F — wa n 'I is an invariant of weight w + n — 1. 



§ 15. The Adjoint of the Transformed Differential Expression. 



Proposition 6, page 19, gives us, for a change of dependent variable 

 or a multiplication of L(u) by </>, a simple relation between the adjoints 

 of the transformed and the original differential expressions. For a 

 change of independent variables we have the following relation : 



Proposition 23. If L(u) and its adjoint M(v) go over, under a 



change of independent variables, into L(u) and M(v) respectively, 

 then — j— and — —■ are adjoint. To obtain the adjoint of the 



transformed differential expression we have, then, to subject M(y) 

 to the following transformations: 



