Sir W. Rowan Hamilton on Quaternions. 119 



conical locus of a will contain the given vector a'; because if 

 we suppose a = a', we have /3 = 0, and the equation (2.) is sa- 

 tisfied : and in like manner the locus of a contains the vector 

 a v , because the supposition u — ay gives /3" = 0. In the third 

 place, the cone contains a" and « lv ; for if we suppose a = a", 

 then, by the principle contained in the formula (4.) of the last 

 article, we have 



f = - V(V.aV.V.a"a'") = a"S.aV«'"; 

 and by the same principle, under the same condition, 

 V. /3/3' = V ( V (V. &**i Vi x'cc") . V(V. a' a". V. a" a v)) 

 = -V.a'a".S(V.a'"a". V.a'a". V.a'W) . 



but S ( V. a' «" . a") = S. a f a" a" = ; therefore S. (3 ,3' 0" 

 = S (V. j3 /3' . /3") = 0; and in like manner this last condition 

 is satisfied, if a = a lv , because /3 and V. /3' /3" then differ only 

 by scalar coefficients from « 1V and V. a lv a v , respectively, so 

 that the scalar of their product is zero. Finally, the conical 

 locus of a contains also the remaining vector a'", because if 

 we suppose a. = a'", we have 



/3 = a"' S. «' a'" *** /3" = «'" S. a" *'" a*, 

 and therefore in this case S. j3/3'/3" = 0, because the scalar of 

 the product of a'" and (5' a 1 " is zero. The locus of a is there- 

 fore a cone of the second degree, containing the five vectors 

 a', a", «'", a lv , a v ; and in exactly the same manner it may be 

 shown without difficulty that whichever of the six vectors a. .a? 

 may be regarded as the variable vector, its locus assigned by the 

 equation (2.), of the present article, is a cone of the second 

 degree, containing the Jive other vectors. We may therefore 

 say that this equation, 



S./3/3'/3" = 0, 

 when the symbols /3, /3', /3" have the meanings assigned by the 

 definitions (1.), or (substituting for those symbols their values) 

 we may say that the following equation 



S{V(V. a «'.V. a '"a lv ).V(V.«'a".V. a iv a v) < -| 



V(V.aV".V.a v a)} =0, J* ' (3>) 



is the equation of homoconicism, or of uniconality, expressing 

 that, when it is satisfied, one common cone of the second de- 

 gree passes through all the six vectors a a' a" a'" a ,v a v , and 

 enabling us to deduce from it all the properties of this common 

 cone. 



26. The considerations employed in the foregoing article 

 might leave a doubt whether no other cone of the same de- 

 gree could pass through the same six vectors; to remove 

 which doubt, by a method consistent with the spirit of the 



