618 Sir W. Rowan Hamilton on Quaternions, 



plane of the ellipsoid ; and its length is the trigonometric 

 tangent of the angle of rotation in that plane from the direction 

 of the line t to that of the line »'; while a similar interpretation 

 applies to the second member of either of the same two equa- 

 tions, the sign — in that second member signifying here that 

 the two equally long angular motions, or rotations, from t to 

 i', and from t to x', are performed in opposite directions. 

 Thus the vector t, which touches a line of curvature, coincides 

 in direction with the bisector of the angle in the tangent plane 

 between the projections, i' and x', of the cyclic normals there- 

 upon ; or with that other line, at right angles to this last bi- 

 sector, which bisects in like manner the other and supplemen- 

 tary angle in the same tangent plane, between the directions 

 of »' and — x' : since x' may be changed to — x', without alter- 

 ing essentially any one of the four last equations between t, *', x'. 

 Those two rectangular and known directions of the tangents 

 to the lines of curvature at any point of an ellipsoid, which 

 were obtained by the process of article 53^ are therefore ob- 

 tained also by the process of the present article ; which con- 

 ducts, by the help of the geometrical reasoning above indicated, 

 to the following expression for the system of those two tan- 

 gents T, as the symbolical solution (in the language of the pre- 

 sent calculus) of any one of the four last equations (72.),. .(75.): 



t=j!'(U.'±Ux'); (76.) 



where t' is a scalar coefficient. 



The agreement of this symbolical result with that marked 

 (62.) may be made evident by observing that the equations 

 (68.) give 



,' = v-iV.v»; x' = v-iV.vx; . .-. . (77.) 



so that if we establish, as we may, the relation 



^^'=(Tv)-S (78.) 



between the arbitrary scalar coefficients t and t', which enter 

 into the formulae (62.) and (76.), those formulae will coincide 

 with each other. And to show, without introducing geome- 

 trical considerations^ that (for example) the form (73.) of the 

 recent condition relatively to t is symbolically satisfied by the 

 expression (76.), we may remark that this expression, when 

 operated upon according to the general rules of this calculus, 

 gives 



Tx'.V./t=±/'V..'x'; Tx'.S.i'T=^'(-T.i'x'±S..'x');'l ,^^ ^ 

 Ti'.V.Tx'=/'V.i'x'; Ti'.S.Tx'=^'(S..'x' + T.i'x'); J '' 



and that therefore the two members of (73.) do in fact receive. 



