The beautiful result given by Professor Cayley in the Proceedings of the Royal Society (vol. xiii. p. 553), and deduced, as I understand, by the methods of his memoir “On the Conic of Five-pointic Contact” (Philosophical Transactions, vol. cxlix. p. 371), led me to inquire how far the formulæ of my own memoir “On the Contact of Plane Curves” (Philosophical Transactions, vol. clii. p. 41.) were applicable to the solution of the present problem. The formulæ in question are as follows: if U = 0 be the equation of the curve, H its Hessian, and V = ( a, b, c, f, g, h )( x, y, z ) 2 = 0 that of the conic of five-pointic contact; and if, moreover, α, β, γ being arbitrary constants, δ = αx + βy + γz