114 Sir W. Rowan Hamilton on Quaternions. 



therefore the equation (2.) gives also 



aa'" a' = *'*'" a; (4.) 



a result which, when compared with the general equation of 

 coplanarity assigned in the same preceding article, shows that 

 the new vector a!" is coplanar with the two other given vectors, 

 * and *'; it is therefore perpendicular to the vector of their 

 product, V. «a', which is perpendicular to both those given 

 vectors. We have therefore two known vectors, namely 

 V. a. a! and a", to both of which the sought vector a!" is per- 

 pendicular ; it is therefore parallel to, or coaxal with, the 

 vector of the product of the two known vectors last mentioned, 

 or is equal to this vector of their product, multiplied by some 

 scalar coefficient x; so that we may write the transformed 

 expression, 



a '" = xV{V. **'.*") (5.) 



And because the function *"' is, by the equation (2.), homo- 

 geneous of the dimension unity with respect to each sepa- 

 rately of the three vectors * 3 a', a", while the function 

 V(V. **' .*") is likewise homogeneous of the same dimension 

 with respect to each of those three vectors, we see that the 

 scalar coefficient x must be either an entirely constant number, 

 or else a homogeneous function of the dimension zero, with 

 respect to each of the same three vectors; we may therefore 

 assign to these vectors any arbitrary lengths which may most 

 facilitate the determination of this scalar coefficient x. Again, 

 the two expressions (2.) and (5.) both vanish if a" be perpen- 

 dicular to the plane of * and a' ; in order therefore to deter- 

 mine x, we are permitted to suppose that «, a', a" are three 

 coplanar vectors: and, by what was just now remarked, we 

 may suppose their lengths to be each equal to the assumed 

 unit of length. In this manner we are led to seek the value 

 of x in the equation 



xV.aa'.a u = «&.«'«"-«'&«"«, . . (6.) 

 under the conditions 



S.aa'«"=0, (7.) 



and 



a 2 - a '« = «"2 = - 1 ; (8.) 



so that a, a', a" are here three coplanar and imaginary units. 

 Multiplying each member of the equation (6.), as a multiplier, 

 into —a" as a multiplicand, and taking the vector parts of the 

 two products ; observing also that 



V. a' a" = - V. a" a\ and - V. a a" = V. a" a ; 

 we obtain this other equation, 



xV.**' = V.*"*.S.*'*"- V.*"*'.S.*"a; . . (9.) 



