18 Proceedings of the Royal Irish Academy. 



Eeplacing Q by P in (49) and (55) for the sake of greater clearness 

 and choosing the differentials so that S^ is zero, we have 



' P= |S (^+ cr)(d^ + dp) ; Pt-P= -||d^ SSpf ^ + VP 



-JjSYVaYdpSp. (62) 



Comparing the second of these with (60) and observing that 8 and d 

 correspond respectively to d and d', we may write 



cr, = |^+V^; cr3 = VV(7, (63) 



and the conditions (61) are identically satisfied. Now - o- is the 

 vector potential of the magnetic current, and - ^ is the scalar 

 magnetic potential,^ so that if 



p = - E-(T, (64) 



we shall have the integral (60) equal to the difference of the two 

 values of the integral 



P = - jS^d^ (65) 



corresponding to the two modes of passage which together form the 

 limit for the integral (60). The quaternion p may be called the 

 quaternion magnetic potential ; and the magnetic force o-j and the 

 electric displacement — - cr^ are derived from p by the combinatorial 

 operations with Z), 



cr, = -(i),i?), cr3 = [2),i4 (66) 



Art. 15. — "When the integral is not independent of the mode of 

 passage (31) gives 



SQ = ffs(^'- Wo-iVs^Vdpd'p + difVd'pSp + d'^VSpdp) 



-JJSVo-^SSpdpd'p; 

 or supposing the differentials chosen so that dp and ^p are along 



^ Oliver Heavside : Electrical Papers, yol.i.,^. 4:^1 . 



