4 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The Wronskian process for deriving invariants 21 



Every invariant may be expressed as a function of the following 



invariants : b, the numerators of -^ -^— , . . ., and of their 



o o 



derivatives 22 



§ 6. Particular covariants 22-2-1 



§ 7. Multiplication of L(u) by <£; invariants of a differential equa- 

 tion 24-25 



Invariants of L(u) for this transformation are invariants of 



M(v) for v = >//■ • i\ 24 



Definition of an invariant of the differential equation 25 



If 1(a) = J(b) is one, so is J(a) = 1(b) 25 



Definition of the invariant adjoint to a given invariant of the 



differential equation 25 



§ 8. Invariants of the first and second degree of differential expres- 

 sions and equations .' 25-27 



The b's are essentially the only linear invariants of L(u) ... 26 

 Statement of further results 26 



III. Reduction to canonical form 27-39 



§ 9. Ordinary differential expressions 27-30 



Complete system of invariants of L(u) =0 29 



Every invariant is a function of the invariants /„_*.- , I n -k, i 29 



Process for deriving invariants 30 



§ 10. Partial differential expressions ; conditions for the possibility 



of the reduction 30-33 



The property is invariant 30 



Second order 31-32 



nth order 32-33 



§ 11. Partial differential expressions, continuation ; invariants thus 



suggested 34-39 



Results 35 



Examples 37 



Processes for deriving invariants 39 



IV. Change of independent variables; invariants and covariants . . 40-50 

 § 12. General properties 40-42 



Coefficients of transformed differential expression .... 40 



Definition of invariant, covariant ' . . 41 



Every invariant is isobaric 41 



Every im'ariant is the sum of homogeneous invariants ... 42 

 § 13. Particular invariants and covariants 42-45 



A , 2, Ay dxi dxj, Z a { j — -r—, ; for second order 43 



Generalization of the last 44 



Generalization of the invariant -=- 44 



dx 



§ 14. Reduction to canonical form of an ordinary differential ex- 

 pression 45-47 



Results 46 



List of invariants 46 



Process for deriving invariants 47 



