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metrically interpret 5a, 5/3, 5y, by considering these infinitely small 

 variations of a, /3, y, as arising in the passage from the given ray to 

 an infinitely near ray of the system. The plane which passes through 

 the given ray, and is parallel to the infinitely near ray, may be called 

 the plane of osculation : since, if it be known, we shall know the 

 ratios of 5a, 5/3, 5y, and can determine, by the formula (V), the 

 position of the focus of the osculating system. To simplify this 

 determination, let us put 



A' = X, + »R, Y=y, + fiR, Z = z, + yR, (Y') 



X, Y, Z, being the coordinates of the focus, and .r,, y,, z,, having 

 the same meanings as in the eighth number; the formula (U) then 

 becomes, by the nature of TV, and by the relations (G), 



RS.v + i'W= .r ^^ + y,i^ l^ + .,S, |L. ^2') 



d^V denoting 



S«3 





The second number of this equation {Z'), vanishes when the ray 

 passes through the origin ; and if we suppose the ray to coincide 

 with the axis of z, we shall have also Sy = 0, and the equation will 

 become, 



«=(«S - W) - + ^ i"^. + S) '* + ("W + W>'' <^'' 



which expresses the dependence of the parameter R, on the ratio of J/3 

 to dot; R being now the distance from the origin, upon the ray, to the 



focus of the osculating system ; and the ratio jj being the tangent of 



the angle 0, comprised between the plane of xz, and the plane having 

 for equation, 



,, .,.., ,, ^ = r-= tan.?. (B") 



