422 Mr. A. Russell on the Magnetic 



2, Mathematical Formula?. 



It is convenient to collect together for reference the 

 mathematical definitions and theorems in connexion with 

 Elliptic Functions which we shall require in proving the- 

 theorems which follow. Most of tbem are well known, and 

 they are nearly all to be found in Legendre's Traite des 

 Fonctions Elliptiques (1825). For proofs the reader is 

 referred to A. Cayley's ' Elementary Treatise on Elliptic 

 Functions/ 



The definitions of the complete elliptic integrals E and F 



are 



V0 



E 



_j>. „„„ F =n 



where A= (1 — & 2 sin 2 <£) ' , and k is the modulus. 

 It follows at once from the Integral Calculus that 



M{mD<mH)1-(^)'!si 



»-H'+S)*»+tf^(H#*+--'}- 



Hence when U is small E and F can be readily computed. 

 When h is nearly equal to unity (which is its maximum. 

 value) Legendre * has shown that we can use the formulae 



+SrM l0 4-.o"-sTi) + -- (1 > 



where h ] 2 = l—k 2 , and so l\ is a small quantity. For 

 instance, when k = 0'99, ^ = 0'14 approximately. For a 

 proof of (1) and (2) see A. Cayley, 'Elliptic Functions, 7 2nd 

 edition, p. 54. 



* A. le Gendre (A. M. Legendre), " Memoir -e stir les Integrations par 

 arcs d 1 ellipse,'" Histoire de l'Academie Royale des Sciences, 1786. 



