1888.] Theorems in Analytical Geometry. 391 



By means of the two equations — 



= S.-j 



since P, Q, R, S are functions of m and a, we determine the double 

 tangents. 



We may also use the above equations to determine the two tan- 

 gentials of any single tangent of a quartic in the point where the 

 tangent meets the curve again. In this case we assume ra, a, a as 

 known, and we have — 



2a + 7 + 8 = P, 



a 2 + 2* 7 + 2*<3 + 7 a = Q, 



from which the co-ordinates of the tangentials may be determined by 

 the solution of a quadratic equation. 



We next proceed to find the equations which determine the bitan- 

 gents of the quintic. We substitute y = mix + a in the general 

 equation, and obtain (using the same notation) — 



* + /3 + 7 + S + ,a = P, 

 a/3 + *7 + a.8 + a/A -f- /3 7 -f ft8 -f /3/i + 7# -f 7/1 + fy* = Q, 

 a^7 + «/3<> + otfi/i + «7§ + «7^* + <*fy* + ^7^ + /fy/* + /36/i -j- 76//- = R, 

 a/?7^ + a/37/1 + a/3fyi + a7^ytt + /^fyi == S, 

 a/3 7 S/t = T. 



Put a = /3, 7 = 5, then the equations become — 

 20 + 7) + H = Bi 



a 2 + 7 2 + 2(a + 7) /t + 4*7 = Q, 

 2a 2 7 + 2*7 2 + (a 2 + 7 2 ) fi + 4a 7 /i = R, 

 a 2 7 9 + 2a 7 (a + 7) /t = S, a 2 7 2 /t = T. 



Hence a + 7 = — ^ — ' 



(« + 7 + /0 2 + 2a 7 = Q + /i 2 , 



2*7 = Q - T - - 2 ~ + "J • 

 Hence the remaining equations become — 



