Egberts — On the Satellite of a Line meeting a Cubic. 293 



XLYIII. — Ojf THE Satellite of a Line meeting a Cttbic. By 

 "William E. Eobekts, M. A. 



[Eead, April 12, 1880.] 



Befoke entering on the discussion of the equation of the satellite and 

 that of the tangent at the three points in which a given line meets a 

 cubic, it will be found convenient to premise the following theorem : — 



Given the equations of two curves of the p*^' and £*'' order, respec- 

 tively, 



(1). ^{xy^) = 0, 



(2). Hxyz) = 0, 



to form symmetric functions of the pq values which simultaneously 

 satisfy the two equations. Eliminating xy% between (1) and (2), 

 and the equation of an arbitrary line Ix + my -{■ %n = 0, we obtain an 

 equation of the pq^^'' degree in I m n, which may be written 



A(<^, ./.) - Ap„o,ol"' + A,,_„„oP"'-'m + . . . &c,; 



then we shall have — 



■^pqj o> — K . Xi X2 • • ' Xpg ; 

 ■^pq-ij 1)0 = K. .2i yi Xo X'i . . . Xpf^ ; 

 &c., &c. 



Having found these fundamental symmetric functions, the formation 

 of others presents no difficulty. 



Let T = be the equation of the tangents at the points where 

 L ~ Xx + ixy + vz meets the cubic 17= x^ + y^ + %^ + &mxy%, then 

 we must have — 



T = ^U - KBM, 



where S is the condition that L should touch TJ and M E A'« + i>!y + v'z 

 is the satellite of L. Por, when Z touches If, T must reduce to the 

 product Z^Jf multiplied by a numerical factor K. 



The polar conic H of any point x'y'"^ on T, TJ, and Z, all pass 

 through a common point. Hence the eliminant of 8, Z, and ZT", 

 equated to zero, will express that x'lj':^ lies on T. Oi', 



T ^ [x^ + y^ + t.{ + 6mXiyiZi) {xi + y? + t-? + 6«j . x^jjo'^o) ; 



^'i^/i-ij x-iyoZz being values common to S and Z. It will be sufficient 



