5H Sir W. Rowan Hamilton on Quaternions. 



and may regard these two vector parts of these two quater- 

 nions, or of the products vjw- and v», as denoting two equally 

 long straight lines. Consequently the vector +vt, which has 

 the direction of the line represented by the pure vector pro- 

 duct v(|«. + »), or by the sum V.vjw, + V.v< of two equally long 

 vectors, has at the same time the direction of the sum of the 

 two corresponding versors of those vectors, or that of the sum 

 of their vector-uiiits ', so that we may write the equation 



/vT=UV.vjct + UV.vj, (60.) 



where U is (as in art. 19) the characteristic of the operation 

 of taking the versor of a quaternion, or of a vector; and t is 

 a scalar coefficient. Again, the equation 0=S.v/ax, (54.), 

 which expresses that the three vectors v, jix,, x are coplanar, 

 shows also that the two vectors V. vju, and V. vx are parallel to 

 each other, as being both perpendicular to that common plane 

 to which V, /x, and x are parallel ; hence we have the following 

 equation between two versors of vectors, or between two vec- 

 tor-units, 



UV.Vi(;t=±UV.vx; ...._. (61.) 

 and therefore instead of the formula (60.) we may write 



^T = v-iUV.v.±v-aiV.vx (62.) 



In this expression for a vector touching a line of curvature, or 

 parallel to such a tangent, the two terms connected by the 

 sign + are easily seen to denote (on the principles of the 

 present calculus) two equally long vectors, in the directions 

 respectively of the projections of the two cyclic normals » and 

 X on a plane perpendicular to v; that is, on the tangent plane 

 to the ellipsoid at the proposed point, or on any plane parallel 

 thereto. If then we draw two straight lines through the point 

 of contact, bisecting the acute and obtuse angles which will in 

 general be formed at that point by the projections on the tan- 

 gent plane of two indefinite lines drawn through the same 

 point in the directions of the two cyclic normals, or in direc- 

 tions perpendicular to the two planes of circular section of the 

 surface, the two rectangular bisectors of angles, so obtained, "will 

 he the tangents to the two lines of curvature : which very simple 

 construction agrees perfectly with known geometrical results, 

 as will be more clearly seen, when it is slightly transformed as 

 follows. 



54t. If we multiply either of the two tangential vectors t by 

 the normal vector v, the product of these two rectangular vec- 

 tors will be, by one of the fundamental and peculiar^ prlnci- 



* See the author's Letter of October 17, 1843, to John T. Graves, Esq., 

 printed in the Supplementary Number of the Philosophical Magazine for 

 December 1844: in which Letter, the three fundamental symbols i,J,i 

 were what it has been since proposed to name direction-units. 



