118 Sir W. Rowan Hamilton on Quaternions. 



more symmetrically, the equation 



aa + a'a' + a"a" = 0, (8.) 



introducing three scalar coefficients a, a', a", which have how- 

 ever only two arbitrary ratios, as a symbolic transformation 

 of the proposed equation a a! a" — a" «' a = 0. And it is i'e- 

 markable that while we have thus lowered by two units the di- 

 mension of that proposed equation, considered as involving 

 three variable vectors a, a!, a", we have at the same time in- 

 troduced (what may be regarded as) two arbitrary constants, 

 namely the two ratios of a, a', a". A converse process would 

 have served to eliminate two arbitrary constants, such as these 

 two ratios, or the two scalar coefficients b and b' } from a linear 

 equation of the form (8.) or (7.), between three variable vec- 

 tors, at the same time elevating the dimension of the equation 

 by two units, in the passage to the form (2.) or (1.). And the 

 analogy of these two converse transformations to integrations 

 and differentiations of equations will appear still more com- 

 plete, if we attend to the intermediate stage (5.) of either 

 transformation, which is of an intermediate degree, or dimen- 

 sion, and involves one arbitrary constant b ; that is to say, one 

 more than the equation of highest dimension (1.), and one 

 fewer than the equation of lowest dimension (7.)* 



25. As the equation S. ax' a." = has been seen to express 

 that the three vectors a a' a" represent coplanar lines, or that 

 any one of these three lines, for example the line represented 

 by the vector a, is in the plane determined by the other two, 

 when they diverge from a common origin ; so, if we make for 

 abridgement 



^ = V(V.aa'.V.«'"a 1 v), -i 



/3' = V(V.a'a".V.a lv * v ), [ . . . . (1.) 



/3" = V(V.a"a"'.V.a v «), J 

 the equation 



S./3j3'j8" = (2.) 



may easily be shown to express that the six vectors aa! a." a!" a™ a? 

 are homoconic, or represent six edges of one cone of the second 

 degree, if they be supposed to be all drawn from one common 

 origin of vectors. For if we regard the five vectors a' a" a'" a lv a y 

 as given, and the remaining vector a as variable, then first 

 the equation (2.) will give for the locus of this variable vector 

 a, some cone of the second degree; because, by the defini- 

 tions (1.) of /3, j3', /3", if we change a to a a, a being any scalar, 

 each of the two vectors (3 and /3" will also be multiplied by a, 

 while /3' will not be altered ; and therefore the function S./3J3'/3" 

 will be multiplied by a 2 , that is by the square of the scalar a, 

 by which the vector a is multiplied. In the next place, this 



