162 MR. J. J. WALKER ON THE DIAMKTERS OF A PLANE CUBIC. 



, / a< a# ,o4> a<j> a^> ,a</.\ 



\b (n ." m ^ t -r TO ^ mflJ 

 \ oy oz oz ox ox oy/ 



= (cm 2 + bft 2 - 2fmn) /W + (an 2 + cZ 2 - 2gnl) (^V+ (W 2 + am 2 - 

 4- 2 (- amn - ft 2 + glm + hnZ) ^ ^ + 2 (~ b^ + flm - gm 2 + hmn) 



2 ( c?m + fw J + gw 1 ' 1 



the substitution of the values (30), (31) of (d<f>/dx) z , . . . , c<f>/dy 3<f>/3z, . . . , in which, 

 with the addition and subtraction of the terms, wherein L = Ix + my -\- nz, 



a/ 



Lh 



gives an expression identically equal to the form (29). 



25. The cases of the application of this theorem which occur are : (1) When 

 ( 28) <f> is a cubic u, and i/r is the polar line of a point whose coordinates are 

 ndu'/dy' mdu'/dz . . . , (x'y'z') being also a point on Ix + my + nz = > so that 

 a = 3 2 </8a; 2 . . . f = d"<f>/dy 9z . . . ; in which case 



and the quadratic functions in Z, m, n which multiply <f> and \fi respectively become 

 (bc'+ b'c - 2f f ') P + . . . , +2 (gh' 4- g'h - at" - a'f) mn + . . . , and 36*' 



rpesectively, where a' = d z u'/dx'* . . . f ' = d' 2 u'/dy' dz'. . . . 



Now, otherwise, the substitutions may be made in the conies (da/ox) ... of the 

 other form of the polar line, 



