4<26 Sir W. Rowan Hamilton on Quaternions. 



mentioned in article 64, of which the quaternion equations 

 are, by article 62 (Phil. Mag. for July 1848), 



T(p-,.)=T(p-X') = i. (114.) 



72. Here (see Phil. Mag. for May 1848), we have for jtx, 

 the value, 



l«,=^'(x— »)i eq. (91.), art. 57 j 

 and 



a'(x'- »') = x'p + px', eq. ( 1 1 0.), art. 60 ; 

 also 



»x'=»'x = T.»x, eq. (107), same article; 



whence we derive for >! the expression, 



>!='^^^^ = ^:l^. . . . (HO.) 

 » '— X ' * — rx * 



But 



{i-^^K-')-'={i{it-i)K-'}-^ = )i{}i-i)-h-'', . (141.) 



and by (104.), 



,p + p, = -A'(x-02; .... (142.) 

 therefore 



X'=-^'x(x-.)»-i=A'(x-x2,-i). . . (143.) 



If then we make, for abridgement, 



g=-/i'T^, (144.) 



and employ the two new fixed vectors >j and 9, defined by the 

 equations (see Phil. Mag. for May 1849), 



,] = T.U(i-x), 5 = T:<U(x->-»-i), (131.) 

 which have been found to give 



i~x=>,T'-^, x-xVi=-9T*-:=^, (132.) 



we shall have the values, 



(i—gri; A'=g9; (146.) 



and the lately cited equations (114.) of the two sliding spheres 

 will become, 



T(p^gri)=b', T{p-gQ)^b; . . . (146.) 



between which it remains to eliminate the scalar coefficient gf 

 in order to find the equation of the ellipsoid, regarded as the 

 locus of the circle in which the two spheres intersect each 

 other. 



73. Squaring the equations (146.), we find (by the general 



