156 MB. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 



where, x'y'z being a given point on the line, and D defined by 



p(p- 1) . . . (p - r+ 1) D'= {( w - tf)^, + (a - tft + (fi8- ,) , 



if, for shortness, a, ^, y represent the sines of the angles of the triangle of reference, 

 p is equal to the length of the segment of the line between the point x'y'z' and any 

 one of the p points in which it meets the curve u (' London Math. Soc. Proc.,' vol. 9, 

 p. 227) ; provided f 2 + tf + - - 2?? cos A - 2f cos B 2ft cos C = 1. 

 1 7. Thus the sum of the reciprocals of thgse segments is equal to 



p Du/u ; 



while, if (xyz) is any other point on the same line, & + , and p similarly equal to 

 the segment between x'y'z 1 and it, 



-r,a). . . (5) 



But if xyz is the CoxES-point (4) on the line in respect to x'y'z' and the curve u, 

 p/p' = p Du/u 



whence (5) 



(x x') 5-7 + (y y') ~ , -j- (z z') ~-; 

 or, 



viz., the locus of xyz is the polar line of x'y'z. 



18. If now, instead of considering 17, as variable and (x'y'z') as fixed, 77, are 

 regarded as constant say equal to I, in, n and (x'y'z') as any point on L, or 

 Ix + my -f- nz = 0, the envelope of the line (6) will obviously be the same as its 

 equation in line coordinates, regarded as a curve of order p 1 in point coordinates 

 x'y'z'. Thus, if u were a quartic, (3) having been written in the form 



, , ,o , . / / / 



x^ +....+ Zx*y ^ +...+ Gxyz ^^ = 0, 



its envelope might at once be written down as 



&v/3v\2 i . r/av\s9v\* . 



viz., the poloid of the line L and the quartic u is another quartic. I have made the 

 foregoing remark to draw attention to the perfect generality of the theory of the 



