Coefficient of Mutual Induction of a Circle, fyc. 



249 



II - 



de 



(1 - c' sin -d) v 7 !— A-'-siu'V 



The elliptic integral of the third kind II is expressible by means of 

 Legend re's formula* in terms of elliptic integrals of the first and 

 second kind, complete and incomplete, and may so be calculated 

 without difficulty. f 



§ 5. Putting as before 



dM flA da dx 



it is shown that 



a = i^j( F - c ' n) 



where U = — - 2 - +— 2 (F— II) = v M. ' 



§ 6. The mutual potential energy of a circular current and a uni- 

 form coaxial circular cylindrical current sheet (the current lines 

 being in planes at right angles to the axis) is identically the same as 

 the mutual potential energy of the circular current and a coaxial 

 helical current of the same radial and axial dimensions, beginning 

 and ending in the ends of the sheet, if the current across a length of 

 a generating line of the sheet equal to the pitch of the helix is equal 

 to the helical current. 



§ 7. The mutual potential energy of a helical current and a uniform 

 coaxial circular cylindrical current sheet, or of two uniform coaxial 

 circular cylindrical current sheets is expressible in terms of elliptic 

 integrals. 



§ 8. The electromagnetic force between a helical current and a 

 uniform coaxial circular cylindrical current sheet is given by the 

 formula 



F = o /r/ (M 2 -M0 



where 7* = the current in the helix, 



7 = the current a2ro£S unit kngth of a generating line of the 

 sheet, 



* 'Cayley,' "Elliptic Integrals,'' § l f .3. 

 f ' Cayley/ chap. 13. 

 VOL. LXII. T 



