10 RESEARCHES ON II. 



§ 3. Integration of the Differential Equations for the Circular Currents. 



The integrals of formulae (5) can be obtained by means of the elliptical functions 

 of the first and second kinds/ 



In order, however, to reduce them to a more convenient form, we shall suppose 



^ = tang, a, 

 m 



whence we have 



p COS. a — m sin. a = 0. 

 Let us also make 



G) = 2 i// + a, 

 then for the quantity under the radical sign 



R^ + P — 2 B (m COS. o + p sin. a), 

 after this substitution, and the evolution of the sines and cosines of the sum of 

 the arcs, and the reductions of the functions of double arcs to simple ones, we 

 shall have 



R~ + P — 2 JR [m COS. a + p sin. a) + 4c B {m cos. a + p sin. a) sin.^ 4' > 

 in which making again the supposition of 



4" ^ it — <^, 

 and reducing the cosine to the sine, we shall have 



R^ + F + 2B [m cos. a + p sin. a) — i R {m cos. a + p sin. a) sin? ^. 

 Let us now make for the sake of brevity 



(1) R {m COS. a + p sin. a) = li^. 

 (2) i2^ + Z" + 2 i2 (to COS. a + p. sin. a) = i2' + Z^ + 2 Zi^ = ¥. 



,qv .2 4: R (m COS. a + p sin. a) 4:h^ 



^ ^ ^ ~ R^ + i:' + 2R {m COS. a + p. sin. a) ~ IF' 

 and remembering that the two suppositions successively made for u and 4' are 

 equivalent to the one, 



6) = 2T]/ + a = 2(| — ^Wa = 7t — 2<|) + a, 



whence 



cZ cj = — 2 d ^, 

 we shall have the following equation : 



/2 7t /*'* 



d cd _ I — 2 d<p 



(B? + 1;' — 2R (m COS. a + p sin. o)) I / 7i^ ( 1 — c" sin.'' ^y,. 

 From the foregoing supposition we obtain also 



^ Legendre, Traits dea Fonctions Elliptiques, I, pp. 255, 257. 



