JoLY — Integrals depending on a Single Quaternion Variable. 19 



instantaneous curves (so that (\.t and Zt are zero), while d'p is an 

 element of a path of a representative point, this reduces to 



hQ 



d'^fsf^'- Wo-iWspdp-IISVcraSSpdpd'/D. (67) 



The difference between two integrals for different modes of passage is 

 therefore 



Q^ = Q-ldi't JJ So-s Sp dp - JIJ S V 0-2 S 8p dp d'p (68) 



if 



^^^-VVcr, = -o-3. (69) 



In the variation from one mode of passage to the other, the in- 

 stantaneous curve corresponding to a given value of t traces out a 

 surface — the instantaneous surface. The integral l\ S 0-3 Sp dp is 

 the flux through this surface, supposed momentarily fixed, and the 

 time integral of this is J d'^ ^^ S 0-3 Sp dp. In the electro-magnetic 

 illustration 0-3 is the conduction cui^rent when the medium is not a 

 perfect non-conductor. S Vo-j is the electric volume-density. (Clerk 

 Maxwell, Electricity and Magnetism^ Art. 619). 



Art. 16. — Finally for the triple integral (40), we take as an 

 example 



Q = JJJ Si^[d^d'^d"<?], (70) 



or in terms of p and t since 



[d^ d'^ d"(?] = S dp d'p d"p - d;^ Y d'p d"p - ^t Y d"p dp 



-d"^Vdpd'p, (71) 



and since we may choose the differentials so that dt and d't are 

 zero, we have 



Q = - J d"^ JJ S Yi? V dp d'p + JJJ S /? S dp d'p d"p . (72) 



The limits now consist of a closed surface composed of pairs of 

 points of departure and arrival corresponding to prescribed times. 

 We may imagine a sui'face drawn through the representative points 

 to sweep thi'ough the closed surface. This variable instantaneous 

 surface must at every instant cut the limiting siu'face in a definite 

 curve corresponding to that instant, but the shape of the instantaneous 

 surface is otherwise arbitrary until the mode of passage is prescribed. 



