DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 381 



SICT. V. 70. General propositions relating to dependence of mixed covarianta. 

 (cont.) 71. Inference as to covariants which involve only the original of the dependent variables. 

 72, 73. Mixed covariants involving the associate variables. 



74. Aggregate of covarianta. 

 75-78. Limitation on number of identical covariants for qoartic and qnintic. 



79. Symbolical expression for Jacobian derivatives. 

 SECT. VI. 80. Application to equation of second order. 



81. Reduction of cubic to canonical form. 



82. The " adjoint " equation of the cubic. 

 83, 84. Simple integrable forms of the cubic. 



85. Quotient-equation for the cubic. 



86. Primitive of the cubic when two particular solutions of its quotient-equation are 



known. 



87. Case of vanishing priminvariant ; cubic quotient-derivative. 



88. Particular integrable case of cubic. 



89. Primitive of cubic when only one solution of quotient-equation is known. 



90. Special equation in Schwarzian derivatives. 



91. Reduction of quartic to general canonical form. 

 92, 93. Alternative canonical form of the quartic. 



94. Quotient-equation for the quartic. 



95. Primitive of the quotient-equation when three solutions are known. 



96. Quartic quotient-derivative. 



97. Primitive of qnartic when three solutions of its quotient-equation are known. 

 98-100. Case of vanishing invariants. 



101. Solution of quartic indicated when only one solution of its quartic equation is known. 



102. Adjoint-equation of the qnartic. 



103. Self-adjoint associate equation of the qnartic. 



104. Form of this equation when the priminvariant O 3 of the quartic vanishes, with a 



particular example. . 



105. General form of the self-adjoint associate equation. 



106. Determinantal relation among the self-complementary variables. 



107. Verification of the theorem that all the priminvariants of the self-adjoint associate 



equation are invariants of the original qnartic. 



108. Other associate equations. 



109. Theorems relating to the general equation of order n. 

 SKCT. VII. 110. Quotient-derivatives of successive odd orders. 



111. Quotient-derivatives and reciprocants. 

 112, 113. Law of transformation of vanishing quotient-derivatives ; converse not necessarily 



true. 

 114-117. Transformation of quotient-derivatives for homographic variation of independent 



variable. 



118. Illustration of limitation on converse of 112. 



119. Derivation of quotient-derivatives of even order. 



120. The hyper-linear quotient-derivative. 



121. The hyper-quadratic quotient-derivative. 



122. Scheme of quotient-derivatives of successive odd orders. 



123. Relation between derivatives of odd and of even orders. 



SKCT. VIII. 124. Reproduction of canonical form of differential equation for homographio transforma- 

 tion of the independent variable. 



