370 Sir W. Rowan Hamilton ow Quaternions. 



takes easily this shorter form, 



p2 + i2^Af. (93.) 



If now we cut this surface by the system of two planes, 

 parallel respectively to the two cyclic planes (23.) and (25.), 

 and included in the joint equation 



{^-h{^-K)}{i^-h'{H-i)}=0, . . . (94.) 

 which is derived by multiplication from the equations (86.) 

 and (91.), we are conducted to this other equation, 



p'i + h'^ = h{ip + pC)j^h\xp + pK) + hh'{i-Kf; . (95.) 

 which ma}' be put under the form 



-b^={p-hi-h'Ky^-{/i + h'){hi^ + h'K^); . . (96.) 

 or under this other form, 



T{p-^)=r, (97.) 



if we write, for abridgement, 



^ = ht + h'x, (98.) 



and 



r=\^{b''-{h + h')(hi'' + h'K^)}. . . . (99.) 



Any two circular sections of the ellipsoid, parallel to two 

 different cyclic planes, or belonging to two different systems, 

 are therefore contained upon one common sphere (97.), of which 

 the radius r, and the vector of the centre ^, are assigned by 

 these last formulas : which again agrees with the known pro- 

 perties of surfaces of the second order. And the equation of 

 the 7)iea7i sphere which contains the iv/o diametral a.\-\A ciYCula): 

 sections, is seen to reduce itself, in this system of algebraical 

 geometry, to the very simple form * 



|52 + i2 = (100.) 



59. The expressions (86.), (91.), (98.), for \, /x, 0, give 



i=:^ = fzit = ^=i^ =/,+/,'; . . . (101.) 



if then we regard A, ju-, ^ as the vectors of the three corners 

 L, M, N of a plane triangle, and observe that 0, i — x, and — x 

 were seen to be the vectors of the three corners a, b, c of the 

 generating triangle described in our construction of the ellip- 

 soid, we see that the new triangle i,mn is similar to that gene- 

 rating triangle abc, and similarly situated \n one common 

 plane therewith, namely in the plane of the greatest and least 

 axes of the ellipsoid ; the sides lm, mn, nl of the one triangle 

 being parallel and proportional to the sides ab, bc, ca of the 



* Compare article 21, in the Phil. Mag. for July 1846. 



