288 Sir W. Rowan Hamilton on Quaternions. 



mity^ of the tangent plane of the ellipsoid, or of an element of 

 that surface, to the centre, at the end of that semidiameter p 

 from which v is deduced by that equation. 



46. Conceive now that at the extremity of an infinitesimal 

 chord dp or t, we draw another normal to the ellipsoid ; the 

 expression for any arbitrary point on the former normal, that 

 is the symbol for the vector of this point, drawn from the 

 centre of the ellipsoid, or from the origin of the vectors p, is 

 of the form p + nv, where n is an arbitrary scalar ; and in like 

 manner the corresponding expression for an arbitrary point 

 on the latter and infinitely near normal, or for its vector from 

 .the same centre of the ellipsoid, is p + dp + (w + dw)(v + dv), 

 where d« is an arbitrary but infinitesimal scalar, and dv is the 

 differential of the vector of proximity v, which may be found 

 as a function of the differential dp by differentiating the equa- 

 tion (31.), which connects the two vectors v and p themselves. 

 In this manner we find, from (31.), 



(x2-,2)2dv = (,2 + x2)dp + .dpx + xdp.; . . (38.) 



and the condition required for the intersection of the two near 

 normals, or for the existence of a point common to both, is 

 expressed by the formula 



p + dp + (w + d«)(v + dy) = p + wv; . . . (89.) 



which may be more concisely written as follows : 



dp + d.nv=0; (40.) 



or thus : 



dp + ndy + dnv = (41.) 



We can eliminate the two scalar coefficients, n and An, from 

 this last equation, according to the rules of the calculus of 

 quaternions, by the method exemplified in the 24th article of 

 this paper (Phil. Mag., August 1846), or by operating with 

 the characteristic S . vdv, because generally 



S.yft2=o, S.vjttv = 0, 



whatever vectors ft, and v may be ; so that here, 



S . vdvwdv = 0, S . vdvdwv = 0. 



• This name, "vector of proximity," was suggested to the writer by a 

 phraseology of Sir John Herschel's; and the equation (31.), of article 4.5, 

 which determines this vector for the ellipsoid, was one of a few equations 

 which were designed to have been exhibited to the British Association at 

 its meeting in 1846: but were accidentally forwarded at the last moment 

 to CoUingwood, instead of Southampton, and did not come to the hands 

 of the eminent philosopher just mentioned, until it was too late for him to 

 do more than return the paper, with some of those encouraging expressions 

 by which he delights to cheer, as opportunities present themselves, all per- 

 sons whom he conceives to be labouring usefully for science. 



