macfarlane: algebra of physics 401 



+ 2-^-/hj+2-^-/jk + 2-^-/M. 

 dxdy dybz ozox 



But this supposes that h j k are constant. When they are 

 variable we have the additional terms 



dx I dx h dy j dzk ) 



d?/ Idxh dyj dz k) 



dz (dxh by j dzk) 



This formula still applies when the antecedent units have a 

 modulus attached, as in the case of spherical coordinates. 



Axial units. The principle that a parallelogram is equal to 

 the scalar product plus the orthogonal product when applied to 

 two line units a and (3 may be written 



aj8 = COS a/3 . a 2 + sill a/3 . a [a/3] 



7T/2 



a 



By dropping the alphas in the latter term, and supposing the 

 dimensions placed in the ■ axis [a/3] we pass to the corresponding 

 axial unit, and write 



a/3 = COS a/3 . a 2 + sin a/3 . [a/3] 7 "" 72 

 The index t/2 remains, or, what means the same thing, an i 

 is attached to [a/3] as i [a/3]. Consider next the unit a/37. We 

 have 



a/37 = (COS a/3 . a 2 + sin a/3 . i [a/3]} 7 

 = COS a& . a 2 7 + i sin a/3 COS [a/3] 7 . [a/3] 3/2 

 — sin a0 sin [a/3] 7 . [[a/3] 7]. 

 The second term has a scalar unit, but it is different in kind 

 from a 2 ; because [a/3] has two dimensions and the projection of 7 

 along it only one. Hence the unit is [a/3] 3/2 . It also involves an i. 

 The third term reduces to 



- cos 7a . 7 2 /3 + cos py . /3 2 a; hence 



a/37 = COS a/3 . a 2 7 — COS 7a . 7 2 /3 + COS 187 . /3 2 a 

 + sin a/3 COS [a/3] 7 . i\apf /2 



an equation which satisfies the principle of dimensions. 



