MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 163 



and then the multiplier of 3u'/2 (the present value of p<j>l(p 1) in (29)) becomes, 

 for i/ = Zdu/dx, 



or, identically, that multiplier becomes 



a* 7 a^ ,.3*' 



oc i<; - . *H\ 



OO-\ w W % / ' Ov t\f 



ox oy o~ 



for 



OU < '' , ' ' 



3x 3y ~ 3c 



respectively. 



Thus the result of the substitutions becomes 



'' **' . . (32) 



whereas, by making the substitutions in the first way, the result would be 



Identifying these two results, 

 (bc'-f b'c - 2ff')P + . . . + 2 (gh' + g'h - af - a'f)w + . 



so that the polar of x'j/z with respect to the poloid of L may be written 



av ah* , yu' \ /y^ a a 



ay + ay" a^ " ^ 



26. In the second case which occurs of similar substitutions ( 30) 



= 6u 



the doubles of the coefficients of s ; and 



Ix + my -\- nz= L 

 Y 2 



