174 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 



42. But considering a point (x'y'z) on L, the double chord through it is, plainly, 

 that connecting it with the point in which its polar line touches the envelope s, viz., 

 this is the polar with respect to s of the CoTES-point on that polar-line ( 29). Its 

 equation, thus considered, is therefore 



/3s \ /a\ /a\ 



with the coordinates of the CoTES-point referred to substituted for xyz in (ds/dx) 

 (3s/3?/) (3s/3z); those coordinates being (36). 29, 



a/ ;k' _ a^a_^ a' ay 3' a?*' a/ aj</ jv ay 

 a7 37 ~ a/ a/' a^ ^ 7 a** 3a/' s/ 3^' ~ fa W 



The resulting equation to the double chord is, therefore, 



_ 

 a* a*' ay "~ a/ 3 M ~ a* a a? 



, > 



= 



43. This being cubic in x'2/V, and of five dimensions in the coefficients of u, its 

 envelope, obtained as the condition that Ix + my' + nz' = shall touch the ternary 

 cubic in x'y'z', will be a curve of order 4 in xyz, of order 6 in Imn, and of order* 8 in 

 the coefficients of u. 



This envelope, as well as that of the complex of chords of u, connected with the 

 polar line of a point on L by having their CoTES-points on it, but not passing through 

 its pole, will be more conveniently considered by means of special forms of s and u, 

 to which every cubic may be reduced. 



* This would appear to be 20, but another form of the equation of the double-chord, into which the 

 coefficients of u enter only in the second degree, may be obtained from that of s (8 bis) given in the 

 Note to 22. Combining (i) and (iii) of that Note, the coordinates of the point of contact of the 

 polar line of (x'y'z) with the poloid (D, fig. 1) are at once given as proportional to 



respectively ; hence, the line joining this point with (x'y'z') is 



w' (aDw'j + y Dw' 2 + z DM' S ) DM' (xu\ + yu' 3 + zu' s ) = 0. 



But, if 6a . . . 6/ . . . stand for rftuldx- . . . &u/dy 3 . . . ; \, /, v for ^m ftn, n 7?, fil am, 

 respectively, 



DM'! = \a' + ph' + v g', D' s = \h' + f ti + /', Du' s = \g' + /. 



while, since Ix' -f my' + nz' = 



t'ft y'v = z' (an <yl) y' (fil urn) = I (a: + fly' + 72') = ZA'; x'v z'\= mA'; y'\ x'fi= iA'. 



Substituting in (i) the equation of the double-chord (DE, fig. 1) becomes divisible by A', and is 



a'\ , a^u ,a , a\ 



(5b bls) 



(Note added April, 1888.) 



