36 On certain Geometrical Theorems. [May 25, 



, DVD 4 (jcv)-D*M8D 3 (ajv) 



WilGIlCO fit— v ^ y *£ _JLZ 



and D»ylDV-pVD»y 



P D»(«»)D^ -D*(«y):py 



then /t, /9 are found from (1), (2), (3), and the conic determined. 



(5.) Let us now endeavour to show how the equation to the cubic 

 with 9-pointic contact with a given curve is to be found. 



Writing the cubic 



aif'x + bxhj + cip + dy + exyz=y 5 + /j.x* + vx z + px + <r, 



we differentiate eight times and obtain five equations from which v, 

 P, a have disappeared. 



aD\y^x) + &D 4 (a%) + cDPy* + dDhj + eD*(a?y) =D V, 



with four others obtained by substituting in this equation D 5 , D 6 , D 7 , 

 D 8 successively for D 4 . Hence we shall have — 



D^V), D 6 (afy), Dy, Dy D*(ay) | 

 a. 1 D 6 (y 2 a>), D«(afy), Dy, Dy D«(ay) j> 

 D?^), D 7 (a5 y, DV, D7y, D7(a,y) 

 D^), D8(afy), D8yS, Dfy D8(^)J 



fD 5 (^), Dy, Dy D 5 (a^)1 

 V 1 D7(^), D7^ } D 7 y , D 7(^) F 



lx> 8 (aty), "Dy, D 8 i/, D 8 (ry j 

 fD*(a*y), Dy, Dy D*(ay) 1 



+ T>5 bJ D6 (^)' D V* Dy D»(ay) I 

 ' y j D7(^), D 7 y8j D 7y, D7(,y f 



lB\xhj), Dy, Dy D*to)J 



fi^W, Dy, Dy D^n 



4. T)6 3 D5 (^)' D V. D ^ D5 ^) L 

 * D7(^) 5 Wy\ D7y, D7(.^) [ 



[ IP&y), Dy, I)8y, D8(«y) J 

 DVW, DV, Dy D*(*y) 1 



.to , i D 5 (^y, Dy, Dy D%y ! 



S D^y, Dy D*(«y) f 



L DB(^), Dy, B% B%vy) J 



