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II. group of the group of the transformations T. and of weight w. / g dx I \_ V "T |" , I I I , ) J so that (25) is seen to be a covariant for the sub-group W= if C7" is 4. Canonical form of the differential equation and of From we eery, a covariant of degree & and of weight w. (21), its invariants. , find w = *r rA + -si -Pi i . = ^-, " i 1? , , where ^ (26) whence According to (9) we shall therefore find P is ^- (27) Being a seminvariant, 2 not changed by any transformation = ly y affecting only the dependent variable.

He found the semin variants essentially by historical nature. In 1862, Cockle started a series , method which we have adopted, well as the semi-covariants, without proving, however, the completeness of the system. He also found one function, invariant under transformations of the independent the as of the alone. In 1879 Laguerre^) found the invariant 3 d equation of the 3 order, and showed that its vanishing is the condition for a homogeneous quadratic relation between its solutions, 3 a result which we shall verify later.

We Pp 2 . . , place that, if the formulae expressing be derived from (21), these are linear in k \ observe in the first P terms of P* in and the , explicitly, others "determined Are these terms also expressible as integral Pn and of their derivatives? 3 P P rational functions of form contain, beside which have in their uncanonical invariants linear those terms which have been Aki are algebraic functions of the derivatives of |. The invariant equations could clearly be obtained by eliminating these It must therefore be possible, derivatives of | from the equations.