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the two sides Bc and ac of the triangle, while «— « will be 
one value of the variable vector p or Ea, namely, the remain- 
ing side of the same triangle, or the semi-diameter Ba in the 
last mentioned construction of the surface; and by applying 
to this equation (44) the general methods which the author 
has established for investigating by quaternions the tangent 
planes and curvatures of surfaces, it is found that the vector 
of proximity v of the tangent plane to the centre of the ellip- 
soid (that is, the reciprocal of the perpendicular let fall on this 
plane from this centre), is determined in length and in di- 
rection by the equation, 
(x? — ?) vy = (2 + &) p + ipk + pe; (45) 
while the two rectangular directions of a vector 7, tangential 
to a line of curvature, at the extremity of the vector p, are de- 
termined by the system of equations : 
vetrv=O03~) vrirk — krivv — 0; (46) 
which may also be thus written : 
S.wr(h:.. 5). stirk= 0: (47) 
Of these two equations (46) or (47), the former expresses 
merely that the tangential vector + is perpendicular to the 
normal vector y; while the latter is found to express that the 
tangent to either line of curvature of an ellipsoid is equally 
inclined to the two traces of the planes of circular section 
on the tangent plane, and therefore bisects one pair of the 
angles formed by the two circular sections themselves, which 
pass through the given point of contact. Indeed, it is easy to 
prove this relation of bisection otherwise, not only for the 
ellipsoid, but for the hyperboloids, by considering the common 
sphere which contains the circular sections last mentioned ; 
the author believes that the result has been given in one of 
the excellent geometrical works of M.Chasles; it may also 
be derived without difficulty from principles stated in the mas- 
