307 



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



point of contact ; the equation of mi 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) + id^fip) ; 



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



= 2 S . vdp + S . dvdp, 

 if 



df(p) = 2s:vdp. 



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

 represented by the equation 



Ap dp 



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



osculation, and 



„ c?v _ S.dvdp _ dy{p) 

 Tp^ ~dp^ ~ 2d^ 



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

 direction of osculation. Hence the important formula : 



= o -7-; 



p — <r^ ap 



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

 lates in the direction answering to JJdp. 



" By combining this with the expression formerly given 

 by me for a normal to the ellipsoid, namely 



(k^ - l^y V = (l^ + K^) p + t/OK + Kpi, 



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

 surface may easily be obtained. And for any 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 opp', which is drawn through a given normal po, and 



