Mr. Maxwell on Faraday's Xmes of Force, 319, 



We haye seen that from certain expressions for magnetic intensity 

 we could deduce those for the quantity of currents, so that the cur- 

 rents which pass through a given closed curve may be measured by 

 the total magnetic intensity round that curve. Here we have an 

 integration round the curve zV^e/^ instead of one over the enclosed sur^ 

 face. In the same way, if we assume the mathematical existence of 

 a state, bearing the same relation to magnetic quantity that mag- 

 netic intensity bears to electric quantity, we shall have an expression 

 for the quantity of magnetic induction passing through a closed cir- 

 cuit in terms of quantities depending on the circuit itself, and not 

 on the enclosed space. 



Let us therefore assume three functions oixy z, O'of^oyo, such that 

 «i 61 Ci being the resolved parts of magnetic quantity, 



dz dy ' dx dz ' dy dx' 



then it will appear that if we assume -~, — V^, -~- as the expres- 



dt dt dt 



sions for the electromotive forces at any point in the conductor, the 

 total electromotive force in any circuit will be the same as that ex- 

 pressed by Faraday's law. Now as we know nothing of these in- 

 ductive effects except in closed circuits, these expressions, which are 

 true for closed currents, cannot be inconsistent with known phaeno- 

 mena, and may possibly be the symbolic representative of a real law 

 of nature. Such a law was suspected by Faraday from the first, 

 although, for want of direct experimental evidence, he abandoned his 

 first conjecture of the existence of a new state or condition of matter. 

 As, however, we have now shown that this state, as described by 

 him (Exp. Res. (60.)), has at least a mathematical significance, we 

 shall use it in mathematical investigations, and we shall call the 

 three functions ao, /3o, 70. the electrotonic functions (see Faraday's 

 Exp. Res. 60. 231. 242. 1114. 1661. 1729. 3172. 3269.). 



That these functions are otherwise important may be shown from 

 the fact, that we can express the potential of any closed current by 

 the integral 



J(«.««J+*A|+<'.yo|)*. 



and generally that of any system of currents in a conducting mass 

 by the integral 



m 



(aoffg + /3o&2 + 70^2) ^ ^y dz- 



The method of employing these functions is exemplified in the 

 case of a hollow conducting sphere revolving in a uniform magnetic 

 field (see Faraday's Exp. Res. (160.)), and in that of a closed wire 

 in the neighbourhood of another in which a variable current is kept 

 up, and several general theorems relating to these functions are. 

 proved. M'^PH'ik 



