Sir W. Rowan Hamilton on Quaternions. 327 



Thus if, in particular, the six vectors a..a v are such as to 

 satisfy the condition 



S.j3j3'j8"=0, .'■ (5.) 



they will satisfy also this other condition, or this other form 

 of the same condition : 



S.aa!a" 8.a!'a!"a lv S.au.'oC S.a v a /f/ « iv . . 



S^Wa" ' S.a"a'a" == S^^'a^ * ~S^V*^ 5 " * ^ > 

 and reciprocally the former of these two conditions will be 

 satisfied if the latter be so. 



These two equations, (5) and (6.), express, therefore, each 

 in its own way, the existence of one and the same geometrical 

 relation between the six vectors a. a! a" a 1 " a lv a v : and a slight 

 study of the forms of these equations suffices to render evident 

 that they both agree in expressing that these six vectors are 

 homoconic, in the sense of the 25th article; or in other words, 

 that the six vectors are sides (or edges) of one common cone 

 of the second degree. Indeed the equation (5.) of the present 

 article, in virtue of the definitions (3.), coincides with the 

 equation (2.) of the article just cited, the symbols /3, /3', /3" re- 

 taining in the one the significations which they had received 

 in the other. The recent transformations show, therefore, 

 that the equation of homoconicism, assigned in article 25, may 

 be put under the form (6.) of the present article, which is dif- 

 ferent, and in some respects simpler. The former expresses a 

 graphic property, or relation between directions, namely that 

 the three lines j3, /3', |3", which are the respective intersections of 

 the three pairs of planes (otal, a'"a lv ), (a' a!', u lv a. y ), (a." a!", a? a), 

 are all situated in one common plane, if the six homoconic 

 vectors be supposed to diverge from one common origin; the 

 latter expresses the metric property, or relation between mag- 

 nitudes, that the ratio compounded of the two ratios of the two 

 pyramids (a a' a") («"a'"a lv ) to the two other pyramids (ota!"a") 

 (a." a! a™), or that the product of the volumes of the first pair 

 of pyramids divided by the product of the volumes of the 

 second pair, does not vary, when the vector a", which is the 

 common edge of these four pyramids, is changed to the new 

 but homoconic vector a v , as their new common edge, the four 

 remaining homoconic and coinitial edges a«'a'"« lv of the py- 

 ramids being supposed to undergo no alteration. The one is 

 an expression of the property of the mystic hexagram of Pas- 

 cal ; the other is an expression of the constancy of the anhar- 

 monic ratio of Chasles*. The calculus of Quaternions (or the 



* Although the foregoing process of calculation, and generally the method 

 of treating geometrical problems by quaternions, which has been extended 

 by the writer to questions of dynamics and thermology, appears to him to be 



