JoLY — Integrals depending on a Single Quaternion Variable. 17 



point. "We may therefore conceive a curve connecting the series 

 of representative points to sweep across the closed curve in a manner 

 prescribed by the mode of passage. Let us suppose the differentials 

 chosen so that dp is an element of this instantaneous ciu've while 

 d'p is an element of the path of a representative point. "We shall 

 then have d^ = 0, and the integral becomes 



Q = - J d7 J S 0-, dp + JJ S 0-, dp d'p (60) 



and -JSo-idp is now (compare Art. 11) the flow of the vectors o-i 

 along the instantaneous curve from one extremity to the other. In 

 like manner J S 0-2 dp d'p is the flux of the vectors cr^ through the 

 elementary strip between two consecutive instantaneous ciirves, the 

 integration being performed along an instantaneous curve ; but for 

 the reason stated in Art. 11, JJSo-gdpd'p is not the flux of the 

 vectors 0-2 through the surface generated by the instantaneous curve, 

 being rather the integral of the fluxes at successive intervals of 

 time through the strips determined by successive positions of the 

 instantaneous curve. 



Art. 13. — We have seen (36) that the conditions that this integral 

 should be independent of the mode of passage are 



SVcr2 = 0, ^^ = VVo-i. (61) 



l^ow these are precisely the equations which the electric displacement 



[-— o-2| in a dielectric and the corresponding magnetic force (o-j) 



satisfy. It is therefore possible to give a physical illustration of the 

 integral (60). Any closed curve being taken in the dielectric, if a 

 variable curve is subject to the conditions that its exti'emities shall 

 move in a determinate manner along the fixed cui-ve ; then the time 

 integral of the flows of the magnetic force from one extremity of the 

 variable curve to the other in every position of the curve added to 

 477 times the integral of the displacement through the strips between 

 successive positions of the variable curve, is independent of the nature 

 of the variable curve. 



Art. 14. — When the double integral is independent of the mode of 

 passage, it may be expressed as the difference of two single integrals. 



K.I.A. PROC, A'OL. XXIV., SEC. A.] B 



