Mil. J. J. WALKER OX THE DIAMETERS OF A PLANE CUBIC. 



157 



poloid, and, at the same tune, to show that it is of equal or higher degree than u for 

 values of p exceeding 3. 



19. In the case, then, alone contemplated in this Memoir, of u being a cubic, and 

 (6) being, therefore, otherwise written 



I3 &u 



a^" 1 



the envelope takes the form (1) 



as given by SALMON ('Higher Plane Curves,' 3rd edition, p. 15G), /, m, n here 

 replacing a, ft, y. 



If written in the normal form of a conic the coefficients w n . . . ?< 23 . . are the contra- 

 variant conies of the triad of conies, 



-o 



w '=a,/ 3 > 



a/ 

 ^ 3 ' 



(7) 



taken singly and in pairs, viz., the cubic being written as in the ' Higher Plane 

 Curves,' only with the substitution of e for m, 



cz 8 + 

 and the poloid of u and L being 



s = + 

 then 



n = (b l c l e 2 )? + (ac, O 



2 + njgyz + ?< 31 2x -f 



-f 6ezyz, (8) 

 (9) 



= (063 c 2 2 )P + (ca 3 c 



ae)mn 

 + 2(a 2 e a^b^nl + 2(a 3 e 



6 1 2 )/r + 2(6^ a 2 6 3 )m 

 + 2(/>,6 3 be)nl + 2(6 3 e 6,c 2 )Znj 



e 2 )n 2 + 2(0^ c 2 a s )wn 



-j- 2(c 2 e tgC^nZ + 2(c 1 c 2 ce)lm 



ca 2 + C 2 a 3 2c,e)m 2 -f (ba 3 -j- a 2 6 3 26 1 e)t 2 

 + t 2(b l c l + e 8 a 2 c 2 a s b 3 )mn-\- ^(b^ bc^nl-^- 2(6 3 C! cb^l 



= (cfc, -r- tgC! 2c 2 e)/ 2 + (ac a^m- + (a& 3 + r/ 3 6, 2a^e)n z 



+ 2(a 2 rj ac 2 )7nn + 2(a 2 c 2 -f c 2 ftjCj a 3 b s )jil -{- 2(c 2 8 ca 2 )/ 



= (^G! + ijCj 26 3 e)/ 2 -f (ac s + a 2 Cj 2aye)m 2 + (at a 2 fe,)/r 



+ 2(a 3 6 1 abjmn + 2(/> 3 a 8 ba 3 )nl + 2(0,63 -f e 2 c^ ^iC 



(10) 



