380 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOT1ENT- 



every concomitant can be expressed as an algebraical function of the concomitants 

 which have already been obtained, and that their aggregate is therefore functionally 

 complete. 



The following is a tabular index of the contents of the paragraphs of the Memoir: 



SECT. I. 1-3. Analogy between transformations of algebraical equations and of differential equations. 



4. Results of MALET and COCKLE on sem in variants of differential equations. 



5. Invariants given by LAGUERRE and BRIOSCHI. 



6, 7. HALPHEN'S results in his memoirs of 1880 and 1883. 



8. Difference of canonical form chosen by HALPHEN from that in this memoir. 

 9, 10. Statement of the aggregate of invariantive functions. 

 SECT. II. 11, 12. General transformation of a linear differential equation. 



13. The invariant due to LAGUERRE and BRIOSCHI. 



14. Definition of index of invariant. 



15-17. Dimension-number and homogeneity of invariants. 



18. Proper invariants. 



19, 20. Method of infinitesimal variation; modification of relations of 11. 

 21-25. The invariants 6 3 , 6 4 , 9 6 , e g , 6 7 . 



26. Number of non-composite linear independent invariants. 

 27, 28. Form of the linear invariants. 

 29, 30. Canonical form of differential equation. 



31. General value of Q a for this canonical form. 



32. Summary of results of section. 



SECT. III. 33, 34. Quadrinvariants ; the quadriderivative process. 

 35, 36. Jacobians ; aggregate of proper qnadrinvariants. 



37. Invariants of the qnartic. 

 38-40. Proper cubinvariants. 

 4143 ; 44. Proper quartinvariants. 



45. Proposition relating to Jacobians. 



46. General conclusion as to quartinvariants. 



47. Proper quintinvariants. 



48. General propositions. 



49. Aggregate of derived invariants. 



50. Semi-canonical form of invariants. 



51. Finality of results. 



SECT. IV. 52. LAGRANGE'S " equation adjointe " of the equation of order n. 

 53, 54. Sets of variables subject to same linear transformation. 



55. CLEBSCH'S theorem on classes of variables in algebraical quantics. 



56. Application of CLEBSCH'S theorem to sets of variables. 



57, 58. Selection of those variables which are also subject to functional transformation. 



59. Dependent variables associate with original dependent variable. 



60. Digression on invariants. 



61, 62. Inferences as to the variables and the linear equations satisfied by them. 

 SECT. V. 63-65. Derived identical covariants in the original variable. 



66. Limitation on their number when associated with a differential equation. 



67. Restricted limitation on their number when associated with a differential quantic. 



68. Derived identical covariants in the associate variables. 



69. Mixed Jacobians. 



