440 Mr. A- Russell on the Magnetic 



and using (3)^ we get 



M=2f(a + %*[l(F-E)+ 1 ^(F-II)], . (3») 

 where 



Jo 



(1 - c 2 sin 2 d) ( 1 - /j 2 sin 2 0) V» 



and is consequently the complete elliptic integral of the third 

 kind in which n= — c 2 . This is the final form in which Jones 

 gave his formula. 



If we put -\\r-7r, and _p = 0, so that e=&, then, by (6), 

 we get 



M = 4tt VS-{(2/A-*}F-(2/*)E}, 



which is formula (20). 



M may be found from (35) by a formula, due to Legendre, 

 given on p. 138 of Vol. I. of his Traite des Fonetions Ellip* 

 tiques, and by the tables given in Vol. II. 



Legendre's formula may be written in the form 



¥-U = JW -^ [E . F(k r , «) + F . E(*', «) 



k z sin a cos a L \ i \ > 



-F.F(£»-7r/2]. . (36)* 



In this formula k'' 2 —l — P, and F(&', a), E(&', a) represent 

 elliptic integrals of modulus k' and amplitude a, where 



Sinarrrfl-C 2 ) 1 / 2 /^'. 



We shall now find the mutual inductance between a helix 

 and a coaxial concentric cylindrical current sheet. Let N t 

 be the number of turns of the helix, 2Jii its axial length, and 

 a its radius. If p be the pitch of the helix, h 1 =pN 1 7r. Let 

 also N 2 , 2/i 2 , and b be the number of turns, axial length and 

 radius of the cylindrical sheet, Let us consider a filament 

 of the cylinder at a height z above the median plane. Then 

 its distances from the ends of the axis of the helix will be 

 /*! + ;£ and i — z respectively. Hence if dM. be the linkages 

 of the flux due to unit current in the helix embraced by the 

 circular filament N 2 (cfe/27< 2 ), we get by (33) 



M ___ ±abc 2 r^ sin 2 6>eos 2 6> f 7 h +z 



a p J l-c j, sin s ^L{(Ai+i) 8 +"a?} , / a 



, . hvzl 1 ,7/? N ^~ 



* See also Cayley's 'Elliptic Functions/ Chapter V. p. 139, 2nd edition. 



