SirW. Rowan Hamilton on Quaternions, 515 



pies of the calculus of quaternions, a third vector rectangular 

 to both ; we shall therefore only pass by this multiplication, 

 so far as directions are concerned, from one to the other of the 

 tangents of the two lines of curvature: consequently we may 

 omit the factor v~* in the second member of (62.), at least if 

 we change (for greater facility of comparison of the results 

 among themselves) the ambiguous sign + to its opposite. 

 We may also suppress the scalar coefficient i, if we only wish 

 to form an expression for a line t which shall have the required 

 direction of a tangent, without obliging the length of this line 

 T to take any previously chosen value. The formula for the 

 system of the two tangents to the two lines of curvature thus 

 takes the simplified form : 



T=UV.v» + UV.vx; (63.) 



in which the two terms connected by the sign + are two vec- 

 tor-units, in the respective directions of the traces of the two 

 cyclic planes upon the tangent plane. The tangents to the 

 two lines of curvature at any point of the surface of an ellipsoid 

 (and the same result holds good also for other surfaces of the 

 second order), are therefore parallel to the two rectangular 

 straight lines which bisect the angles between those traces ; or 

 they are themselves the bisectors of the angles made at the 

 point of contact by the traces of planes parallel to the two 

 cyclic planes. The discovery of this remarkable geometrical 

 theorem appears to be due to M. Chasles. It is only brought 

 forward here for the sake of the process by which it has been 

 above deduced (and by which the writer was in fact led to 

 perceive the theorem before he was aware that it was already 

 known), through an application of the method of quaternions, 

 and as a corollary from the geometrical construction of the 

 ellipsoid itself to which that method conducted him*. For 

 that new geometrical constructioii has been shown (in a recent 

 Number of this Magazine) to admit of being easily retranslated 

 into that quaternion form of the equation f of the ellipsoid, 

 namely 



T(ip + px)=x2— ,2^ equation (9.), art. (38.), 



as an interpretation of which equation it had been assigned by 

 the present writer; and then a general method for investiga- 

 ting by quaternions the directions of the lines of curvature on 

 any curved surface whatever, conducts, as has been shown (in 



* See the Numbers of the Philosophical Magazine for June, September, 

 and October 1847; or the Proceedings of the Royal Irish Academy for 

 July 1846. , 



t Another very simple construction, derived from the same quaternion 

 equation, and serving to generate, by a moving sphere, a system of two 

 reciprocal ellipsoids, will be given in an early Number of this Magazine, 



2 L2 



