Newson: geometry of one dimension. 113 



X and y from J(V"W) and the first polar of (Xj, y,) with respect to 



dV dV d\\^ dW 



either V or W; 1. e., x, - +y,-, =0, or x, -, ly.—, — =0. Or, 



^ dx -^ Vly ^ dx -^ ^ dy 



what amounts to the same thing, by eliminating Xj and y, from 



J(VjWj) and the polar points of (x^, y^) with respect to either \' 



,,. . dV, dV, dW, , dVV^ 



or W: 1. e., X— -^ + y-— -^---0, or x - — ^^ + V-; — ^=0. 

 dXj tly^ dx^ dyj 



When the two quantics V and \V are of the same degree //, they 

 determine an involution of the //th order. The function M for the 

 groups V and W of this involution is the same as for any other two 

 groups V and W. For M(VW) is defined as the locus of the polar 

 points of the Jacobian of V and W with respect to either V or W. 

 If now instead of W we take another group W belonging to the same 

 involution, the Jacobian of V and W is the same as that of V and 

 W; and consequently M(VW') is the same as iM(VW). Again if we 

 take another group V instead of V, it may be shown in the same 

 way that M(V'W') is the same as M(VVV'). Hence we may speak 

 of the function M for an involution just as we speak of the Jacobian 

 of an involution. 



It has been shown that when \V is of the first degree the Jacobian 

 of V and W is the first polar of W with respect to V. It is also 

 known from the theory of poles and polars that, if p,, p^, p^, . . .. 

 Pu— 1 ^s the ;/ — I first polars of \V' with respect to V, then W is the 

 polar point with respect to V of each of the points pj, p._,, . . . .p;,_i. 

 Hence when W is of the first degree, there is no group M(V\V) other 

 than the point W. 



When the two quantics V and W are so related that W is the Hes- 

 sian of V, there is a group of points M(V) of the same order as the 

 Jacobian of V, which is the locus of all polar points of JfV) with 

 respect to V. (This group of points M(V) is an important covariant 

 of V; but it is not, like J(V), a fundamental covariant of V.) 



Heretofore we have implicitly assumed that \' and W were both non- 

 singular quantics, i. e., that neither of them had a multiple point; (a 

 multiple element is the only singularity which a one dimensional geo- 

 metric form can have). We shall now proceed to examine the func- 

 tions J(VW) and M(VW) when one or both the quantics have a 

 multiple point. 



Suppose that V has a multiple point of order /i, then this same 

 point will appear as a multiple point of order (k — i) on each of the 

 first polars of V. Consequently it will appear as a multiple point of 

 order {k — i) on the Jacobian of V and W. This may also be shown 

 analytically by choosing the multiple point for one of the ground 



