336 macfarlane: algebra of physics 



It was shown that a quaternion could be reduced to the sum of 

 two homogeneous components, by writing 



a a a = a Jcos a . a + sin a . a 7 " 72 }; 



and it was pointed out that the righthand member was not a 

 full* equivalent of the lefthand member, as all the complete turns 

 had been dropped. 



It was shown that the fundamental principle in the composition 

 of quaternions is 



f* 7 »/i = _ C os fiy - sin 07 \fiy] r/i = [fr]' +<fiy 



This principle makes both terms negative, in contrast to the 

 Hamiltonian principle which makes the latter term positive, in 

 order apparently that the angle may be less than t. When both 

 terms are negative, it is much easier to pass to the analogous 

 principle for unit vectors. The principle is proved as follows: 



Choose the standard case where the angle between the planes is 

 less than tt/2 (fig. 2). The order from to 7 (by short way) 

 B together with the righthanded screw 



determine the pole [0 7 |. Then 7r/2 T x/2 

 is the resultant of the quadrants AB 

 and BC. The positive angle round OB 

 which will take OA into the position OC 

 is not the angle which takes the shortest 

 way but the angle which is determined 

 by the screw rule, namely ir + < 07 : 

 hence 



f/&. 2. „7T/2 TT/2 n n Tn 1*72 



^ (3 7 ' = — COS 07 — Sill 07 [07j 



According to Gibbs the dyad AL is a kind of symbolic product 

 in which A is the antecedent and L the consequent. I pointed 

 out that according to the use which he makes of the dyad, it is 

 not a product but a quotient, and expresses the ratio of the con- 

 sequent to the antecedent. And in the expression for V it seemed 



. 1 du 1 



plain that to express a rate, d x u -j—r or -j~ 7 was more correct 



du 

 than 3- h. 

 dx 



