286 
of GEOMETRICAL MULTIPLICATION (and division), which were 
embodied by him (in 1843) in his fundamental formule for the 
symbolic squares and products of the three coordinate charac- 
teristics (or algebraically imaginary units) ¢, 7,k, which entered 
into his original expression of a QUATERNION (w+?“-+jy +h2), 
and by which he succeeded in representing, symmetrically, 
that is, without any selection of one direction as eminent, the 
three dimensions of space. 
It is, however, convenient, in many researches, to retain 
the notation in which Greek letters denote vectors, instead of 
employing that other notation, in which capital letters (a few 
characteristics excepted), denote points. In the former nota- 
tion it was shown to the Academy in last December (see 
formula (21) of the abstract of the author’s communication of 
that date), that the equation of an ellipsoid, with three unequal 
axes, referred to its centre as the origin of vectors, may be put 
under the form : 
(ap + pa)’ — (Bp — pf) = 15" 
p being the variable vector of the ellipsoid, and (3 and a being 
two constant vectors, in the directions respectively of the axes 
of one of the two circumscribed cylinders of revolution, and 
of a normal to the plane of the corresponding ellipse of con- 
tact. Decomposing the first member of that equation of an 
ellipsoid into two factors of the first degree, or writing the 
equation as follows: 
(ap + pa+ Bp — pf3) (ap + pa— Bp + pB) =1, (32) 
we may observe that these two factors, which are thus sepa- 
rately linear with respect to the variable vector p, are at the 
same time conjugate quaternions ; if we call two quaternions, 
Q and KQ, CONJUGATE, when they have equal scalars but 
have opposite vectors, so that generally, 
* Appendix, No. V., page lviii. 
