' macfarlane: algebra of physics 371 



The paper then proceeds to vector differentiation, by which 

 is meant the theory of v". It was pointed out that the differen- 

 tial of a reduced expression is not equivalent to the differential 

 of the primitive expression; and that if the axis p is variable, 

 the reduction p 2 = 1 can be introduced only after the process of 

 differentiation has been completed. 



In the investigation of V it is shown that 



VR = 3; v> = 1/p; and V P = 2/r. 



The next step is to investigate d(l/p)/dp; it is found to be not 



; but — . The general conclusion was reached that when n 



p- p- 



is odd 



d (1/p-O/dp = n i^+i ; 

 p 



but when n is even, a minus is introduced. The latter statement 

 was afterward found to be erroneous; in no case is there a minus. 

 In this respect the differential of an inverse power of a unit differs 

 from the differential of an inverse power of a modulus. 



The rest of the paper was devoted to deriving V 2 from V by 

 direct operations of the calculus. For 



\-* /k +JL/j + ±/k, 



ox by bz 



it was sought to deduce V 2 by means of the multiplication formula 



\bx by bz I \bx by bz I 



the result was 



V* = — / h? + — If + — /k 2 , 

 bx 2 by 2 bz 2 



which reduces to 



d 2 d 2 b 2 

 bx 2 by 2 bz 2 



As the above is the formula for conjugate multiplication, it is 

 evident that what was obtained is the conjugate square of V. 



