Sir W. Rowan Hamilton on Quaternions, 289 



In this manner we find from (4<l.) the following very simple 



formula: S.vdvd/3=0; (42.) 



which is easily seen, on the same principles, to hold good, as 

 the quaternion form of the differential equation of the lines of 

 curvature on a curved surface generally ^ if v be still the vector 

 qfproximity of the siiperjicial dement of the curved surface to 

 the origin of the vectors p, which vector v is determined by 



the general condition S.vdo = 0, (43.) 



combined with the equation already written, 



S.vp=l (S*.); 

 or simply if v be a normal vector, satisfying the condition (43.) 

 alone. Substituting, therefore, in the case of the ellipsoid, 

 the expression for dv given by (38.), and observing that 

 S . vdp^ = 0, we find that we may write the equation of the lines 

 of curvature for this particular surface as follows : 



S.v(»d/5x+xdp«)dp = 0; .... (44.) 



which equation, when treated by the rules of the present cal- 

 culus, admits of being in many ways symbolically transformed, 

 and may also, with little difficulty, be translated into geome- 

 trical enunciations. 



47. Thus if we observe that, by article 20, <tx — xt< is a 

 scaiar form, whatever three vectors may be denoted by (, x, t; 

 and if we attend to the equation (43.), which expresses that 

 the normal v is perpendicular to the linear element, or infini- 

 tesimal chord, dp; we shall perceive that, for every direction 

 of that element, the following equation holds good: 



S.v(»dpx— xdpj)dp = (45.) 



We have therefore, from (44.), for those 'particular directions 

 which belong to the lines of curvature, this simplified equation; 



S.v»dpxdp = 0; (46.) 



which may be still a little abridged, by writing instead of dp 

 the symbol t of a tangential vector, already used in (36.) ; for 

 thus we obtain the formula : 



S.V.TXT = . (47.) 



We might also have observed that by the same article 20 

 (Phil. Mag., July 1846), jtx + jctj and therefore <dpx + xdpj is 

 a vector form, and that by article 26 (Phil. Mag., August 

 1846), three vector- factors under the characteristic S may be 

 in any manner transposed, with only a change (at most) in the 

 positive or negative sign of the resulting scalar ; from which 

 it would have followed, by a process exactly similar to the 

 foregoing, that the equation (44.) of the lines of curvature on 

 an ellipsoid may be thus written, 



S.vdp.dpx = 0; (48.) 



Phil, Mag. S. 3. Vol. 31. No. 208. Oct. 1847. U 



