116 Sir W. Rowan Hamilton on Quaternions. 



we have therefore this other expression, after interchanging, 

 as is allowed, the places of the two vector factors a' a" of a 

 binary product under the characteristic S, 



«'" = v(«s.«"«'-« , s.«"«). . . . (i.y 



Substituting here for the characteristic S, that which is, by 

 article 18, symbolically equivalent thereto, namely the charac- 

 teristic 1 — V, and observing that 



= V (a a" ei - a! a" a), (2.)' 



because, by article 20, a. a." a!— a' a" a. is a scalar form, we ob- 

 tain this other expression, 



a'"= s V(a'V.«"«-«V.a"«'). > • • (3.)' 

 The expression (1.)' may be written under the form 



«'" = V(«S.« , « I '-«'S.««"); . . . (4.)' 

 and (3.)' under the form 



a '"=V(aV.a'a"-«'V.aa") ) .... (5.)' 

 obtained by interchanging the places of two vector-factors in 

 each of two binary products under the sign V, and by then 

 changing the signs of those two products; taking then the 

 semisum of these two forms (4.)', (5.)', and using the symbolic 

 relation of article 18, S + V = l, we find 



a W = _Lv( a a'a"-«' a «") 



35 



= vQ-(aa'-a'a).a"); .... (6.)' 



in which, by article 20, —(aa' — a' a) = V. a a! ; we have 



s 



therefore finally 



a'" = V(V.«a'.a"): (7.)' 



that is, we are conducted by this purely symbolical process, 

 from laws of combination previously established, to the trans- 

 formed expression (12.) of the last article. 



24-. A relation of the form (4.), art. 22, that is an equation 

 between the two ternary 'products of three vectors taken in two 

 different and opposite orders, or an evanescence of the scalar 

 part of such a ternary product, may (and in fact does) present 

 itself in several researches ; and although we know, by art. 

 21, the geometrical interpretation of such a symbolic relation 

 between three vector forms, namely that it is the condition of 

 their representing three coplanar lines, which interpretation 

 may suggest a transformation of one of them, as a linear func- 

 tion with scalar coefficients, of the two other vectors, because 

 any one straight line in any given plane may be treated as the 



