Sir W. Rowan Hamilton on Quaiernions, 3G9 



(82.), we employ this other general formulaj which likewise 

 holds good for any three vectors, 



,p+px=(p+^)(;c-.), . . . (87.) 



we shall thereby transform the lately cited equation (9.) of the 

 ellipsoid into this other form, 



t(p+!^')=*; (88.) 



which is analogous to the form (8.3.), and from which similar 

 inferences may be drawn. Thus, we may treat this equation 

 (88.) as the result of elimination of a new auxiliary vector 

 symbol fj. between the two equations, 



^(j- x) = if) + ^j; (89.) 



T(p-/^)=i; (90.) 



of which the former, namely the equation (89.), is, relatively 

 to p, the equation of a iicw plaiie, parallel to that other cvclic 

 plane of the ellipsoid for which we have seen that 



j/5 + pi = 0, (25.), art. 44; 



while the latter equation, namely (90.), is that of a new sphere, 

 with the same radius h as before, but with ^u, for the vector of 

 its centre: which sphere (90.) determines, by its intersection 

 with the plane (89.), a 7ie-ia circle as the locus of the termina- 

 tion of/3, when /a receives any given value of the form 



[t=h'{K-,), (9].) 



where h' is a new scalar coefficient. The ellipsoid (9.) is 

 therefore the locus of all the circles of this second system al.-o, 

 answering to the equations (89.), (90.), as it was seen to be 

 the locus of all those of the first system, represented bv the 

 equations (84.), (85.) ; which agrees with the known proper- 

 ties of the surface. 



58. For any three vectors », h, p, we have (because p% x% 

 and xp + f,x are scalars) the general transformations, 



{ip+pi)ixp + px) = i{xp+px)p + p{xp + p)i)t 1 



= {iK + xi)p'^-\-tpxp + pxpt I. (92.) 



= -{i-xyp^+{,p+px){pi + xp) ; J 



and therefore, with the recent significations of the symbols 

 b. A, /*, expressed by the formulaa (81'.), (84.), (89.), the equa- 

 tion ot the ellipsoid assigned in a foregoing article, namely 

 {ip + px){pi + xp) = {x^-i'^)% (21.), art. 44, 

 Phil. Mag. S. 3. Vol. 82. No. 216. May 1848. 2 B 



