IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 5 



§ 15. The adjoint of the transformed differential expression . . . 47-50 

 V. Conditions for (p • L(u) being (— 1)" times its adjoint 50-60 



§ 16. The conditions 50-55 



The property is invariant 50 



Ordinary differential expressions of the second order; Sturm's 



Normal Form 51 



Ordinary differential expressions of the nth order 51 



Partial differential expressions of the second order .... 52-53 



Solution of problem for this case 53 



Partial differential expressions of the nth order 53-55 



§ 17. The covariant 2 ( 3 - J '■ - xr ) <*»&& 55-60 



v ( d _k _ ^A 



r. \dxj dxi) 



Particular case : two independent variables 59 



List of invariants and covariants 59 



The following paper deals with linear differential expressions, both 

 ordinary and partial, and of all orders. The term "differential expres- 

 sion," as used in these pages, refers, then, always to linear expressions. 

 After an introduction devoted to the theory of the adjoint differential 

 expression, the invariants and covariants of a differential expression 

 under the three transformations which leave its general form unchanged 

 are considered. 



The presentation of the introductory matter (I) is, in the main, a re- 

 production of the substance of lectures by Professor Bocher in Harvard 

 University, or an extension to expressions of the wth order of matters 

 discussed in those lectures for the second order. The same remark 

 applies to a good part of §§ 4, 5, 7. Acknowledgment of other indebted- 

 ness is made in the text. References to Wilczynski are to his Projec- 

 tive Differential Geometry. The name of Lie might be expected to 

 occur more often in a paper on such a subject ; it is, however, in ob- 

 taining the results recorded in §8 only that I have made use of his 

 methods. 



For permission to use the matter referred to above, as well as for 

 most helpful guidance and suggestion in the preparation of this paper 

 throughout, my warmest thanks are due to Professor Bocher. 



I. The Adjoint Differential Expression. 



§ 1. Ordinary Differential Expressions. 



The first part of this paper deals with the theory of the adjoint 

 differential expression. Let us begin by recalling briefly the facts in the 

 case of an ordinary linear differential expression of the nth order. For 

 details, reference may be made to Darboux, Surfaces, book iv, chapter 



