398 



macfarlane: algebra of physics 



Quaternions and vector-analysis are reconciled and unified by 

 the complementary principles 



(ia) (i(3) = — cos ap — sin a (3 . i [a/3] 

 and 



a/3 = COS a/3 + sin a/3 . I [a/3]. 



Both are unified with algebra and the Cartesian analysis by 

 the development which is given of the line complex of vectors, 

 and the cyclic complex of vectors. 



Line complex of vectors. Let A, B, C, D be a number of line- 

 vectors of which B is applied at the end of 

 i, C at the end of B, D at the end of C 

 (fig. 8). Let this complex be denoted by 5! 

 and the common resultant by R. As the 

 complex is a broken line with its parts in 

 a definite order, but the resultant merely a 

 straight line from the initial point to the 

 final point, the powers of the complex are 

 very different from those of the resultant. 

 Nothing but confusion results from not dis- 

 . tinguishing between the complex and the 



' ' j- Q - resultant. We have seen this already in 



expanding a complex logarithm. 



Let ft = A + B + C + D. The square of ft is the algebraic 

 square of the quadrinomial, with the proviso that the natural 

 order of the vectors be preserved in each term; that is A prior to 

 B, B prior to C, C prior to D. Hence 

 ft 2 = A 2 + B 2 + C 2 + D 2 



+ 2 lAB+AC + AD 

 + BC + BD 

 + CD] 

 and the same is true for any number of terms, 

 the scalar terms be denoted by Cos ft 2 ; then 

 Cos ft 2 = A 2 + B 2 + C 2 + D 2 



+ 2Cos {AB + AC + AD} 



+ 2 Cos {BC + BD} 

 + 2 Cos CD 

 which gives the square of the resultant. 



Let the sum of 



