286 



of GEOMETRICAL MULTIPLICATION (uiid division), which were 

 embodied by him (in 1843) in his fundamental formulae for the 

 symbolic squares and products of the three coordinate charac- 

 teristics (or algebraically imaginary units) i,j, k,vih\c\i entered 

 into his original expression of a quaternion (w + za; 4-7^ + ^2;), 

 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 : 



(„p + ^a)._(/3p_p^)2:^l;* 



p being the variable vector of the ellipsoid, and j3 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 -h /3|0 - p^) (ap +pa-(5p + pjS) = 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 lyiii. 



