Sir W. Rowan Hamilton 071 Quaternions, 4«31 



But a quaternion, such as is u'a or «'"«", is always equal to the 

 sum of its own scalar and vector parts ; and the product of a 

 scalar and a vector is a vector, while the scalar of a vector is 

 zero : therefore 



a'«=S.a'« + V.a'a, «'"«"= S.a"'«" + V. «'"«", . (176.) 

 and 



S . ec"'a"u'ct = S . a"'u". S.u'u + S(V. «"'a". V. a'«). (177.) 



Comparing then (175.) and (177.), and observing that 



S.«a'=+S,«'«, V.aa'=-V.a'a, . . (178.) 



we obtain the following general expression for the scalar part 

 of the product of the vectors of any two binary products of 

 vectors : 



S(V. «'"«".¥. a'«)= S . «'"«. S . «'a"-S . u"W. S.a"« ; (179.) 

 while the vector part of the same product of vectors is easily 

 found, by similar processes, to admit of being expressed in 

 either of the two following ways (compare equation (3.) of 

 article 24) : 

 V(V. a"'a".V.«'a) =a"'S . a"a'a-a"S . u"'u'a 



= aS.a"'a"a'-a'S.a"'a"«; . . . (180.) 



of which the combination conducts to the following general 

 expression for any fourth vector a'", or p, in terms of any 

 three given vectors a, a', a", which are not parallel to any one 

 common plane (compare equation {4t.) of article 26): 



pS.cc"u'u = uS.u"u'p + u'S.u"fcc + u"S.pa'a, . (181.) 



If we further suppose that 



«"=:V.«'«, (182.) 



we shall have 



S.a"«'a=(V.«'«)2=«"2; . . . (183.) 



and after dividing by a"% the recent equation (181.) will be- 

 come 



whereby an arbitrary vector p may be expressed, in terms of 

 any two given vectors a, a', which are not parallel to any com- 

 mon line, and of a third vector u", perpendicular to both of 

 them. And if, on the other hand, we change «, a', «", «'" to 

 e, p, p, >), in the general formula (179.), we find that generally, 

 for any three vectors »j, 9, p, the following identity holds good: 

 S(V.)jp.V.p9)=p2S.»ifl-S.)}p.S.p9; . (185.) 



which serves to connect the two last of the expressions (171.)» 

 and enables us to transform either into the other. 



