390 



Mr. W. H. L. Russell. 



[June 21, 



and therefore the line 



gx + mr]Z + wi^y = 

 must pass through the point of their intersection. So also must 

 7]y + m£x + mz!j = 0, 

 £z m^a? + — 0. 



The pole and the intersection of these two straight lines are called 

 by Dr. Salmon corresponding points. When I had proceeded thus 

 far, and had begun to make deductions from these equations, I became 

 acquainted with the existence of a memoir by Professor Cayley on this 

 subject in the ' Phil. Trans.' for 1857. He has there given these 

 equations without proof. I have therefore demonstrated them exactly 

 in the way in which I discovered tbem before I was acquainted with 

 his paper, to which 1 refer for ulterior theorems. 



To determine the double tangents of a quartic. 



Let y — mx + a be the equation to a straight line cutting the 

 quartic. If this value of y be substituted in the quartic, the equation 

 will become 



x* - Par 3 + Qx* - + S = 0, 



so that if «, /3, 7, B be the roots of this equation, we have the following 

 equations : — 



« + & -f 7 + 6 = P, 

 «/3 + * 7 + *B + £7 + (3B + 7 8 = Q, 

 «/37 + a-jB + <x(5B + /3 7 a = R, 

 ap^b — S. 



Then for the bitangents a, = /3, 7 = B, 



2 + 7) = P, 



2« 7 O + 7) = R, 

 *piB = «y = g f 



O P 3 



( a + 7 )2 + 2* 7 = Q, or * 7 = |- - _ , 



R 2 



and therefore _ = S. 



