307 



if dp be the vector drawn to a point of that plane, from the 



point of contact ; the equation of an osculating surface of the 



second order (having complete contact of the second order 



with the proposed surface at the proposed point) may be thus 



written : 



= dfip) + ^dT'fip) ; 



(by the extension of Taylor's series to quaternions) ; or thus, 



= 2S. Wp+ S.dvdp, 

 if 



df(p) = 2S.vdp. 



" The sphere, which osculates in a given direction, may be 

 represented by the equation 



A/9 dp 



where Ap is a chord of the sphere, drawn from the point of 

 osculation, and 



dv_ S.dvdp _ dy{p) 



dp ' dpi ~ 2dp' 



is a scalar function of the versor Udp, which determines the 

 direction of osculation. Hence the important formula : 



V „ rfv 



= o -J-; 



p- a^ ap 



where a is the vector of the centre of the sphere which oscu- 

 lates in the direction answering to Udp. 



" By combining this with the expression formerly given 

 by me for a normal to the ellipsoid, namely 



(k- - L^y V = (r + K-) p + ipK + Kpt, 



the known value of the curvature of a normal section of that 

 surface may easily be obtained. And for ani/ curved surface, 

 the formula will be found to give easily this general theorem, 

 which was perceived by me in 1824; that if, on a normal 

 plane orr, which is drawn through a given normal ro, and 



