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



no longer vanishes. The result (29), 24, thus becomes 



9 2 w 8%, &u 



{ 

 ( 4 2233 



-f 4L2 {(4 



22 W 33 



mn}. . *. ..... (34) 



It is to be observed that the values of d\jj/dx . . . are to be obtained by substituting 

 9 2 M/3x 2 ... for a ... after differentiation ; thus, generally, in the present case 



= 2ax + 2h/ + 2gz 



27. Lastly, the case which will occur at 38 of the substitutions, 



differs from the first case only in the differential coefficients of u being multiplied by 

 x', y', z instead of x, y, z, so that (29) becomes simply equal to us' ; i.e., us 

 with x'y'z' for xyz. 



III. GENERAL FORMULAE AND EQUATIONS. 



28. The polar line of a point (x'y'z'} on L, or Ix -f- my + nz = 0, meets it in a 

 second point, the coordinates of which are 



x : y : z 



oy' 9z / 9z' 9a/ * 9c' 9w' ' 



and if the coordinates of this latter point are substituted for xyz in the quadric 

 forms (9w/9a;), (9w/9y), (9w/9z) of 



/9w\ 



the polar line ot this point will be expressed in terms of the coordinates of the 

 original point (x'y'z'). 



