400 macfarlane: algebra of physics 



Again Sin R 2 = 2 Sin AB + 2 Sin BC + 2 Sin CA that is, 

 four times the directed areas of the three faces; the resultant of 

 which is equal and opposite to the directed area of the fourth 

 face ABC. 



The Cartesian coordinates involve a cyclic complex. For 

 example R = xh + yj + zk; the square of which is 



R2 = ^2 + y2J2 + g2&2 



+ 2 xy . hj + 2 yz . jk + 2 zx . kh. 

 Application to dyadic $. A dyadic is a sum of dyad quotients 

 LI A + M/B + N/C, of which the antecedents A, B, C, are 

 cyclic, aS also the consequents L, M, N. Hence the dyadic is a 

 cyclic complex of dyads, and may be treated as a cyclic complex. 

 For example, the direct square of $ (not the one commonly treated 

 which is formed after the conjugate multiplication) is 



$ 2 = L-/A 2 + M 2 /B 2 + N 2 /C 2 



+ 2 LM/AB + 2MN/BC + 2NL/CA. 



Hence Cos $ 2 = L 2 /A 2 + M 2 /B 2 + N 2 /C 2 



+ 2Cos LM /Cos AB +2 Cos MN /Cos BC -f 2 Cos NL/Cos CA 



And | Sin * 2 = Sin Liif /Sin AB + Sin ilfiV/Sin EC 

 + Sin NL/S'm (L4, and this last is the invariant commonly des- 

 ignated by $ 2 . 



Application to delta. Suppose / to be a function of r, 6, <p. 

 Then V/ may be defined as the cyclic complex formed of the quo- 

 tients of simultaneous differentials, namely 



V/ = b r f/b r R + b e f/b d R + bJ/b^R; 

 = b r f/br P + bj/r^be + bj/r^ b<p 



bf , , bf , dp . bf , dp 

 = —/p + — r — + — r — . 

 br bd bd dip b<p 



Hence this expression is simply a dyadic in differentials. 



The complete square of V. According to the result obtained 

 in the previous paper 



b 2 b 2 b 2 



v *=°-/h 2 + --/j 2 +—/k 2 

 bx 2 by 2 bz 2 



