512 Sir W. Rowan Hamilton on Quaierm'ons. 



it is now proposed to point out some of the modes of combining, 

 transforming, and interpreting the system of these two equa- 

 tions, consistently with the principles and rules of the Calculus 

 of Quaternions, from which the equations themselves have been 

 derived. 



52. Whatever two vectors may be denoted by t and t, the 

 ternary product tit is always a vector forrn^ because (by 

 article 20) its scalar part is zero ; and on the other hand the 

 square t^ is a pure scalar : therefore we may always write 



T*T = /AT% Ti = |XT, (52.) 



where jw. is a new vector, expressible in terms of » and t as 

 follows : 



H* = T<T-'; (53.) 



so that it is, in general, by the principles of articles 40, 41, 

 42, 43, the reflexion of the vector « with respect to the vector 

 T, and that thus the direction of t is exactly intermediate be- 

 tween the directions of < and jw,. In the present question, this 

 new vector ]«,, defined by the equation (53.), may therefore 

 represent the reflexion of the first cyclic normal «, with re- 

 spect to any reflecting line which is parallel to the vector t, 

 which latter vector is tangential to one of the curves of cur- 

 vature on the ellipsoid. Substituting for m its value (52.), in 

 the lately cited equation (49.), and suppressing the scalar 

 factor T% we find this new equation : 



S.v/*x=0; (54.) 



which, in virtue of the general £'g'Mfl!//ow o/'co/'/awanVj/ assigned 

 in the 21st article (Phil. Mag. for July 1846), expresses that 

 the reflected vector jt*, the normal vector v, and the second 

 cyclic normal x, are parallel to one common plane. This result 

 gives already a characteristical geometric property of the lines 

 of curvature on an ellipsoid, from which the directions of those 

 curved lines, or of their tangents (t), can generally be assigned, 

 at any given point upon the surface, when the direction of the 

 normal (v) at that point, and those of the two cyclic normals 

 (« and x), are known. For it shows that if a straight linejw, be 

 found, in any plane parallel to the given lines v and x, such 

 that the bisector t of the angle between this line ju, and a line 

 parallel to the other given line » shall be perpendicular to the 

 given line v, then this bisecting line t will have the sought 

 direction of a tangent to a line of curvature. But it is pos- 

 sible to deduce a geometrical determination, or construction, 

 more simple and direct than this, by carrying the calculation 

 a little further. 



