260] 
ON THE DOUBLE TANGENTS OF A PLANE CURVE. 
205 
31. Hence the general term of 
D'H — -y- D'fTi + &c. 
vanishes except for S = 0, but when S = 0, its value is 
iC{l, 2 + 8, (r + 1 - 8)} X (_)—[,§]<[ r _ s]r-e . 
or observing that R s ° is equal to [r\ r 4- [s]*[r - s] r “*, the value is simply 
(-)—[r]-(l, 2 + s, (r+l-s)), 
that is, we have 
D r H - ^ D r H 1 + &c. 
= (—)"- r [r] r £*{1, 2 + s, (r + 1 - s)}, 
the summation in respect to s extending from s=0 to s = r. In particular, giving to 
r the values n — 2 and n - 1, and attending to the expressions for III and IV, we find 
A 2 (D n ~-H - D n ~-H l +&c. •••) = - [n - 2] n-2 III, 
A 2 ~ D n ~ 3 H l + &c- [n - S] n ~ 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 curve in question is 
D n ~ 2 H - D n ~ 2 H l + &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 III by 
interchanging (X, F, Z) and (x, y, z), and reversing the sign. Hence if, as before, 
«£), 3), &c. denote the values which H, D, &c. assume by this interchange, we have 
A 2 (^3) n - 2 £ - n -~ 3) H-2 #i +&c. ...) = [n - 2] n-2 II, 
A 2 i T) n_3 4p - ^+ &c. ...^ = [n - 3] n-3 I, 
and the identical equation 
a 0 l + r/j II + a n _j III + ctjiIV — 0