372 Sir W. Rowan Hamilton on Qiiaternions. 



and which is terminated at the point n by the plane of the 

 generating triangle, mnst coincide in direction with the normal 

 V to the ellipsoid : of which latter normal the direction may 

 thus be found by a simple geometrical construction, and an 

 expression for it be obtained without the employment of dif- 

 ferentials. But to show that this geometrical result agrees 

 with the symbolical expression already found for v, by means 

 of differentials and quaternions, we have only to substitute, on 

 the one hand, in the expression (98.) for ^, the following values 

 for h and h', derived from (84.), (86.), and from (89.), (91.): 



h^J^P+l!L H'=-^l±^', . . (104.) 

 -('-") —{^--x-Y 



and to observe, on the other hand, that the equation (31.), 

 which has serveil to determine the normal vector of proximity 

 V, may be thus written : 



(x^-j2)S=(<-x)2p + i(xp + px) + Jc(i^ + |:.); . (105.) 



for thus we are conducted, by means of (81.), to the formula: 



^-p^hh; (106.) 



which expresses the agreement of the recent construction with 

 the results that had been previously obtained. 



60. If we introduce two new constant vectors »' and x', con- 

 nected with the two former constant vectors », x, by the equa- 

 tions 



,x' = ,'x = T..x, (107.) 



which give 



,'2 = ,2^ x''^ = x2, .V = X(, . . . (108.) 



then one of the lately cited forms of the equation of the ellip- 

 soid, namely the equation 



T(ip + px) = x'-»2 (9.), art. 38, 



takes easily, by the rules of this calculus, the new but analo- 

 gous form : 



T(.'pffx')=x'2-,'2 (109.) 



The perfect similarity of these two forms, (9.) and (109.), 

 renders it evident that all the conclusions which have been 

 deduced from the one form can, with suitable and easy modi- 

 fications, be deduced from the other also. Thus if we still 

 regard the centre a as the origin of vectors, and treat i' — x' 

 and — x' as the vectors of two new fixed points b' and c', we 

 may consider ab'c' as a neiio generating triangle, and may de- 

 rive from it the same ellipsoid as before, by a geometrical pro- 

 cess of generation or construction, which is similar in all re- 

 spects to the process already assigned. (See the Numbers of 

 the Philosophical Magazine for June, September, and October, 



J 



