290 ME. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SUEFACES. 
§ 1. Conditions for a Sextactic Point. 
Let U = 0 be the equation to the given surface, and Y=0 that to the surface whose 
section by the plane -uj—Ictb' — 0 is to have a six-pointic contact with the corresponding 
section of U at the point ( x , y, z. , t). Also, following the method of Professor Cayley 
(Philosophical Transactions, vol. cxlix. p. 371, and vol. civ. p. 545), let the coordinates 
of a point of U be considered as functions of a single parameter ; then for the present 
purpose the coordinates of a point consecutive to (x, y, z, t) may be taken to be 
x dx -\-\d 2 x -\--^d 3 x x ^\jyd b x, y-\-dy-\~. . , z-\-dz - (-. . , t-\-dt -\~. . ; . (1) 
and these values when substituted in U must satisfy the equation U = 0. Then writing 
for shortness 
b , = dx cb-f dy b,, + dz b „+' dt b f , 
b 2 —c l 2 xd x -\-d ~yb y + d 2 zb z -\-d 2 tb t , / 9 \ 
y 
b 5 = d 5 xb x T dfdy -j- d/zb , + d 5 tb t ; 
substituting the values (1) in U, expanding as far as terms of the fifth degree, and 
arranging the result in lines of the degrees 0, 1, . . 5, respectively, we shall have 
0 = U 
+b 1 U 
+ i(^i , + 5b 1 b 2 +^3)U 
-^4 ( b ‘ + Gb ;b 2 + 4b jb 3 -f- b J U 
+xio(^x + 10b?b a +10bJb3-l-15biba + 5b 1 b 4 +10b a b3+b 6 )U, y 
each line of which, being of an order different from the rest, must separately vanish. 
Let us write, as usual, 
bjj=?c , b^U = y, b jr U=w, b^U^#,' 
b*U=«„ bl\J = v y , b*U = w„ b?U=&„ 
b,b,u=< b,b,u=p , b,b y u=w', ^ ^ 
bAU =H 9 b,b,U=m', b.b,U=n^ 
Then combining the equation b 1 U = 0 with the corresponding expression in V, viz. 
^¥ = 0, we obtain the usual expressions for two-pointic contact, viz. 
b,V_b y V__b,V _b<V. 
u v io k ' z 
which, since U and V are both homogeneous in x, y, z , t, are equivalent to only two 
independent conditions. These conditions may be comprised in the single formula 
