JIB. A. CAYLEY ON THE DOUBLE TANGENTS OF A PLANE CURVE. 
211 
31. Hence the general term of 
vanishes except for B=0, but when S=0, its value is 
2 +s, (r + 1 — 5)} x(— 
or obser\ing that R® is equal to [r]’‘H-[s]''[r— s]’’"®, the value is simply 
2+5, (r + 1— s)}, 
that is, we have 
D'H-^D^H.+&c. 
=(— )”"W S*{1, 2+5, (r+1— 5 )}, 
the summation in respect to 5 extending from 5=0 to s=r. In particular, giving to r 
the values n—2 and 1, and attending to the expressions for III and IV, we find 
A2^D"-^H-^D”-^H.+&c. ..) = -[?i-2]”-^III, 
A2 +&C. • •) = - [w- 3]”-® IV. 
32. The equation III=0 belongs to the curve which by its intersections with the 
tangent, gives the tangentials of a point of the curve U=0. Hence the equation of the 
cui^ e in question is 
r)n-2H_!i^D«-2H,+&c. = 0, 
which is Mr. Salmon’s theorem, leading to the solution of the problem of double 
tangents. 
33. The expressions for I and II are obtained from those of IV and HI by inter- 
changing (X, Y, Z) and (x, y, z), and reversing the sign. Hence if, as before, H, 29, &c. 
denote the values which H, D, &c. assume by this interchange, we have 
A* + &c. . . ) = [w - 2]"-^!, 
29”-®H, +&C. . .) = [w- 3]«-H, 
1^0 + Ilf^^i + III<5!„_ 1 + IV (l„ = 0 
;^,(d-H-+D*-H, + &c.)®T 
(d”-®h-'^D”-®h,+&c.)t 
and the identical equation 
becomes therefore 
1 = 0 , 
