Sir W. Rowan Hamilton on Quaternions. 115 



in which the three vectors V. u" oc, V. a" a', V. eta! are coaxal, 

 being each perpendicular to the common plane of the three 

 vectors «, «', u" ; they bear therefore scalar ratios to each 

 other, and are proportional (by the last article) to the areas 

 of the parallelograms under the three pairs of unit-vectors, a" 

 and a, a" and a', and a and a', respectively ; that is, to the 

 sines of the angles a, a', and a' — a, if a be the rotation from 

 a," to a, and a' the rotation from a" to a', in the common plane 

 of these three vectors. At the same time we have (by the 

 principles of the same article) the expressions : 



— S. a." « = cos a ; S. a" a' = — cos a' ; 



so that the equation (9.) reduces itself to the following very 

 simple form, 



x sin (a 1 — a) = sin a' cos a — sin a cos a\ . . (10.) 



and gives immediately 



x = 1 (11.) 



Such then is the value of the coefficient x in the transformed 

 expression (5.); and by comparing this expression with the 

 proposed form (1.), we find that we may write, for any three 

 vectors, a, a', a", not necessarily subject to any conditions such 

 as those of being equal in length and coplanar in direction 

 (since those conditions were not used in discovering the form 

 (5.), but only in determining the value (11.),) the following 

 general transformation : 



« S. a' a" -a'S.a"a = V(V. « «' . a") ; . . . (12.) 



which will be found to have extensive applications. 



23. But although it is possible thus to employ geometrical 

 considerations, in order to suggest and even to demonstrate the 

 validity of many general transformations, yet it is always de- 

 sirable to know how to obtain the same symbolic results, from 

 the laws of combination of the symbols', nor ought the calculus 

 of quaternions to be regarded as complete, till all such equiva- 

 lences of form can be deduced from such symbolic laws, by 

 the fewest and simplest principles. In the example of the 

 foregoing article, the symbolic transformation may be effected 

 in the following way. 



When a scalar form is multiplied by a vector form, or a 

 vector by a scalar, the product is a vector form ; and the sum 

 or difference of two such vector forms is itself a vector form; 

 therefore the expression (1.) of the last article is a vector 

 form, and may be equated as such to a small Greek letter; or 

 in other words, the equation (2.) is allowed. But every vector 

 form is equal to its own vector part, or undergoes no change 

 of signification when it is operated on bv the characteristic Vj 



12 



