290 



the two sides bc and ac of the triangle, while t — k 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 isj the reciprocal of the perpendicular let fall on this 

 plane from this centre), is determined in length and in di- 

 rection by the equation, 



(k' - ly v = {k^ + r') p + ipK + Kpi ; (45) 



while the two rectangular directions of a vector r, tangential 

 to a line of curvature, at the extremity of the vector p, are de- 

 termined by the system of equations : 



vT -\- T V :^ ; vTiTK — KTiTv — ; (^^) 



which may also be thus written : 



s . vr = ; s . vTiTK = 0. (47) 



Of these two equations (46) or (47), the former expresses 

 merely that the tangential vector r is perpendicular to the 

 normal vector v ; 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- 



