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through any linear element pp' of the surface, we project the 

 infinitely near normal p'o', which is erected to the same sur- 

 face at the end of the element pp' ; the projection of the near 

 normal will cross the given normal in the centre o of the 

 sphere which osculates to the given surface at the given point 

 p, in the direction of the given element pp'. 

 " I am able to shew that the formula 



= gS^, 

 dp 



which follows from the above, for determining the directions 

 of osculation of the greatest and least osculating spheres, 

 agrees with my formerly published formula, 



= S . vdvdp, 



for the directions of the lines of curvature. 



" And I can deduce Gauss's ^ewera/ properties of geodetic 

 lines by showing that if cri, (T2 be the two extreme values of the 

 vector 0-, then 



= measure of curvature of surface = 



(jO - CTi) (p - 0-2) i^iiJg 



(PTSp 

 ~TSp.dp^' 



where d answers to motion along a normal section, and 8 to 

 the passage from one near (normal) section to another ; while 

 S, T, and U, are the characteristics of the operations of 

 taking the scalar, tensor, and versor of a quaternion : and the 

 variation Sy of the inclination t? of a given geodetic line to a 

 variable normal section, obtained by passing from one such 

 section to a near one, without changing the geodetic line, is 

 expressed by the analogous formula, 



dTdp „ 

 ^'—Tdp' 



