450 Prof. A. Gray on the Dynamical 



that used by Maxwell, which is the method I think generally 

 pursued. Starting from equation (3), viz., 



T=i(N 1 0i + H*&+ ... +»*&.+ . . .), 



take any circuit, say that through which the induction is 

 Ni, and draw any surface in the field so as to form a cap with 

 that circuit as bounding edge. Then, by the well-known 

 theorem as to the work done in carrying a unit pole in a 

 closed path round a circuit, we have 



■yic-- 



: JH ft COS 6 l 



where the integration is taken round any closed path em- 

 bracing the circuit, H k is the magnetic force due to the current 

 in that circuit, and 0k the angle which H& makes with ds, an 

 element of the path. Hence, since the total induction through 

 every such surface is the same, we have 



^kVk= -^ \ N4H* cos Oicds. . . . (15) 



Now let the surface be taken at right angles to the lines of 

 induction everywhere. These lines are closed curves round 

 the conductors, and each threads through one or more of the 

 circuits. It is possible to divide up the whole field by suc- 

 cessive surfaces, each having for bounding edge any given 

 circuit, so that every one of those surfaces shall be everywhere 

 at right angles to the lines of induction. Every one of these, 

 if it cut through a system of closed lines belonging to any one 

 circuit, will pass through every point of that circuit. Of 

 course no one of the closed tubes of induction which the 

 surface thus cuts through contributes anything to the total 

 induction through the surface. 



Now let the direction of the closed curve, round which the 

 integral of H cos 6 ds is taken, be everywhere at right angles 

 to these surfaces, and let B be the value of the induction at 

 any point where this curve cuts one of these surfaces. Then, 

 if dS be the area of an element of the surface at that point, 

 the induction through it is B<iS. Thus we get 



Nfc2/*= j- l \ BH A cos 0j{. ds dS, 



where one integral is taken over the surface and the other 

 round the closed curve. But this is evidently the same 

 thing as 



