18 KESEARCHES ON IL 



Having thus reduced the integral to elliptical functions, we may proceed to the 

 determination of the components. 



I. In the case of real roots, the supposition 



„ = 90° — 2 1^, 

 and the (5) give 



Sin. 6) = COS. 2 4- = (1 — 2 sm.H) =1 — 2 f ''''■' f 



(£ -V COS- <p 



_ (gg + 1) _ (2 y' + 1) sin.^ p 

 ^ + cos? <p 



^ • o I o • 1 1 2q \^ q^ + 1 . COS. * sin. & 



Cos. a = si7i. 2 4- = 2 sm. tL cos. ■yp = — ^ i-j 5— i T 



q^ + cos.^ ^ 



ah — bm sin. a = ab — hn (±1 , -5 — ■ ' • - ) 



\ q^ + cos.^ ^ ' 



_ {ah — mb) (q"" + 1) — {ab— {2 q' + l)bm) sin."" ^ 



q^ + cos? ^ 



Combining these values with those of T in (6), we have for the three components 



the following values : 



X = 



r= 



Z = 



2.hbn \{ (f + 1) — (2 g" + 1) sin? g^] [g" + cos? ^] d^ 



A' * A ' 



sin. ^ cos. ^ {(f + cos.^ ^) d^ 



(2 

 



eaYf <i ((f 



+ 



1)1 



»7t 



4 hanq 







(2 

 



eaYp^q" {<f 



+ 



1)^ 



»7t 



2k 







A' A 



[(ft5 _ hn) (g^ + 1) — {ab — (2 g^ + 1) bm) sin.^ »] 



\<f + cos.^ ^] d^ 

 ~K' 

 The value of T will vanish at the limits, as appears also by the symmetry of 

 the conductor. The values of X and Z must be reduced to more simple forms by 

 reducing the cosines to sines, and dividing. 

 We shall therefore have, making 2^ + 1 = </, 



/n 

 2hbn (f + 2g^ . . sin.* ^ d^ 



(2ea)>«2^(5'^ + l)| " ' A' ' ~A 







/^7t 



2Jcbn (^' + ffcin^^ 2 cf c^ - {g' + 9) 



(2 eafjpY (2^ + 1) 







2fc^~{q^ + g)-c'<f \ d^ 

 c*A^ /A' 



