THE INVARIANTS OF LINEAR DIFFERENTIAL 



EXPRESSIONS.! 



By Frank Irwin. 



Presented by Maxime BScher, April 8, 1908. Received June 9, 1908. 



Contents. 



I. The adjoint differential expression 5-16 



§ 1. Ordinary differential expressions 5-7 



§ 2. Partial differential expressions of the second order 7-11 



Definition of the adjoint, M(v) 8 



Condition for a multiplier 9 



Formulas for coefficients of adjoint 9 



Lagrange's Identity 10 



If vL{u) - uN(v) = % -r- , N(y) = M(v) 10 



Conditions for L(u) being self-adJQint 10 



Three-term form of Lagrange's Indentity 11 



§ 3. Partial differential expressions of the nth order 12-16 



Definition of the adjoint 13 



Condition for a multiplier 13 



Formulas for coefficients of adjoint 14 



Symmetrical formulas for same 14 



Conditions for L(u) being (— 1)" times its adjoint 15 



Lagrange's Identity 15 



II. Change of dependent variable ; invariants and covariants ; invariants 



of a differential equation 17-27 



§ 4. General properties of invariants and covariants 17-19 



Formulas for coefficients of transformed expression 17 



Definitions of invariant, covariant 18 



Every invariant is homogeneous 18 



Definition of weight 19 



Every invariant is the sum of isobaric invariants 19 



§ 5. Particular invariants 19-22 



Adjoint of transformed is ^ times adjoint 19 



The b's are invariants 20 



They constitute a complete system 20 



1 This paper was accepted in June, 1908, by the Faculty of Arts and Sci- 

 ences of Harvard University in fulfilment of the requirement of a thesis for 

 the degree of Doctor of Philosophy. 



