482 Mr. A. M c Aulay on Quaternions as a 



the vector p —p a to vary. On the other hand, in the expression 

 Iff Vi^O'i)^* (where the numerical snffixes imply, as through- 

 out this paper they will imply, that the y with a suffix only 

 operates on the symbols which have the same suffix) either 

 end of p b — n a might be considered the variable one. Since <f> 

 will in general involve other y' s which of necessity must 

 presuppose p b to be the variable, it is convenient to lay down 

 the rule that for all y's under the integral sign p b is supposed 

 to be the variable. Thus when v crosses the integral sign its 

 sign must be changed, or 



vjff*(^ = -iffvi#(ri>fc- ... (3) 



With one exception the only value of r of equation (3) that 

 will be required below is the scalar u defined by 



u = T-H Pb -p a ) (4) 



It is well known that if q be any quaternion function of the 

 position of a point, 



47n/=y 2 Jff^? (5) 



or 



47r?=-v$jv w #?> 

 which gives by means of equation (2) 



4:Trq= — ^uVvqds + ^^u^qd^ (6) 



Here q may be discontinuous at specified surfaces. 



This is the theorem in potentials spoken of. To show that 

 it is really useful let us apply it to an electrical problem. 



Maxwell's theory of the electromagnetic field is w T ell enough 

 known. Let us denote by the term " the ordinary theory " 

 what is now to be described. In the ordinary theory there is 

 a certain vector connected with an electromagnetic field, 

 called the vector potential. This vector consists of two parts, 

 one depending solely on the magnetism of the field and the 

 other depending solely on the currents of the field. The 

 vector magnetic force at a point also consists of two such 

 parts. On the ordinary theory, the magnetic part of the vector 

 potential A and the magnetic part of the magnetic force H 

 are given in terms of the magnetic moment I per volume by 

 certain equations investigated in the 3rd Part (Magnetism) 

 of Maxwell's ' Electricity and Magnetism/ On the ordinary 

 theory, the second part of each is obtained by assuming that each 

 (closed) elementary current produces terms in A and H that 

 would be produced by the corresponding magnetic shell. 



