MR. J. J. WAI.KKK ON TI1K DIAMKTKHS OF A PLANE CUBIC. 165 



Referring to the general form for the result of such substitutions, (29), 24 in the 

 present case it is (32), 25, 



3u' 3u' . 3u 



the form of which shows that the polar urith respect to the poloid (.s) of any point on L 

 cute the polar line with respect to u of that point in its CoTES-point. 



29. The relation just proved gives at once the coordinates of that CoTES-point in 

 the form 



And, plainly, the Corrvs-point of the polar line of a point on L is the pole (with respect 

 to (s) the poloid of L) of that chord of the pencil through the point on L, which passes 

 through the contact with s of the polar line of that point. (Fig. 1.) 



30. From considerations founded on the relation just established the locus of the 

 CoTES-point (xyz) of the polar line of a point (x'y'z) on L may be at once obtained in a 

 general form, viz., by the elimination of x'y'z' among 



3tt' 3u' Sit' 



or 



3% 



or 



, '"- 



* 



\yf + m ,/ + nz ' o. 

 From the last two 



3s 3* , 3* - - - - , ' < 



= n^ w :*5 n ., :m= t^-' 



3y 3a Sz 3 &r 3y 



and the substitution of these values in the first gives the locus of xyz in the form 

 (34) 26, of six times v, if 



