Sir W. Rowan Hamilton on Qiiaternions. 287 



when combined with the equations (27.)} (28.)> and (31.), 

 that we have also this simple relation : 



S.vp = l (34.) 



Subtracting (27.) from (29.), attending to (30.), changing t 

 to Tt. Ut, where U is, as in article 19, the characteristic of 

 the operation of talcing the verso?- of a quaternion (or of a 

 vector), and dividing by Tt, we find : 



• = -^:^^^:^^=2S.vUr + TT./(Ur). {35.) 



This is a rigorous equation, connecting the length or the 

 tensor Tr, of any chord r of the ellipsoid, drawn from the 

 extremity of the semidiameter p, with the direction of that 

 chord T, or with the versor Ur ; it is therefore only a new 

 form of the equation of the ellipsoid itself, with the origin of 

 vectors removed from the centre to a point upon the surface. 

 If we now conceive the chord t to diminish in length, the 

 term TT./(Ur) of the right-hand member of this equation 

 (35.) tends to become =0, on account of the factor Tt; and 

 therefore the other term 2S . vUt of the same member must 

 tend to the same limit zero. In this way we arrive easily at 

 an equation expressing the ultimate law of the directions of the 

 evanescent chords of the ellipsoid, at the extremity of any given 

 or assumed semidiameter p ; which equation isO = 2S.vUT, 

 or simply, 



0=S.VT, (36.) 



if T be a tangential vector. The vector v is therefore perpen- 

 dicular to all such tangents, or infinitesimal chords of the 

 ellipsoid, at the extremity of the semidiameter p ; and conse- 

 quently it has the directiim of the normal to that surface, at 

 the extremity of that semidiameter. The tangent plane to the 

 same surface at the same point is represented by the equation 

 (34.), if we treat, therein, the normal vector v as constant, and 

 if we regard the symbol p as denoting, in the same equation 

 (34.), a variable vector, drawn from the centre of the ellipsoid 

 to any point upon that tangent plane. This equation (34.) 

 of the tangent plane may be written as follows: 



S.v(p-v-')=0; (37.) 



and under this form it shows easily that the symbol v"' repre- 

 sents, in length and in direction, the perpendicular let fall 

 from the origin of the vectors p, that is from the centre of the 

 ellipsoid, upon the plane which is thus represented by the 

 equation (34.) or (37.); so that the vector v itself, as deter- 

 mined by the equation (31.), may be called ihe vector ofproxi- 



