466 Mr. L. Schwendler on the General Theory 



Further, n\ the force exerted by the coil If on a given mag- 

 netic pole when the unit current passes through the coil, can be 

 expressed as follows : — 



n' = r\/3?* 



where r' is a coefficient depending only on the dimensions and 

 shape of the coil, on the manner of coiling the wire, and on the 

 integral distance of the coil from the magnetic pole acted upon. 

 Thus we have 



S / =E , *yy . w _ E , w , & 



b'+f + d' c'+p l 



Now, supposing the factor W constant f, S' becomes smaller 

 the smaller is. 



In the second part it has been proved quite generally that 6 

 decreases permanently with increasing p' and p", no matter to 

 what special cause the variation of c' is due, whence again it fol- 

 lows that p should be a maximum. 



From the form of p, however, we see that for any given sum 

 b +f+ d, p becomes largest if 



which is ' c the regularity -condition " of the differential method. 



To have S, therefore, for any variation as small as possible we 

 must make f= b + d. Substituting this value of /, we get an 

 expression for S which shows that it has an absolute maximum 

 for b } but no minimum, from which we conclude that b should 

 be made either very much smaller or very much larger than the 

 value which corresponds to a maximum of S ; but no fixed rela- 

 tion between b and d or a can be found. 



In order to prove that b-\~d=f is the solution, we must now 

 how that it also makes D as small as possible. 



But as 



we have only to show that the regularity-condition b + d=f 



* This expression supposes that the thickness of the insulating covering 

 of the wire can be neglected against the diameter of the wire, which is 

 allowable, r' is a constant with respect to b'. 



t That W can be kept constant while & decreases and - * varies and 



b' -\-d' 

 f-\-V-\-d : is constant, it will be clear is possible; for if d' :> 0, the variation 

 of b' + d' may be considered entirely due to a variation of d' equal and op- 

 posite in sign to the variation of/'., If 4=0, then we must consider r' 



variable with b' in order to keep W constant while / varies, which is ad- 



b 

 missible, since the position of the coils has not yet been fixed. 



