1'AKT II. POLAR MAGNETIC PHKNOMKNA AND TEKKELLA EXI'ERIMENTS. CHAP. II. 427 



If we employ Legendre's signs, 



lt T) = I y lT^f"sJn% 

 Jo 



E (k lt T) = I y lT^f"sJn% di (17) 







for instance we have, as can easily be proved (see Legendre's 'Fonctions Elliptiques', Vol. I, p. 70), 



sin 2r 



** sin 2 * 



1 k'\ sin 2r 1 sin 2f 



- ,--,- E(ffi,r) - -4 - = - 5- E (* 1( r) tan-V j= - (19) 



< 1 - *i 2i ** sin 2 * cos^v 2 i * sin^r 



as we can at once put 



sinj/ = ^ 1 . (20) 



An angle such as this must in any case be determined, if Legendre's tables are to be used. 

 We have, then 



' 8(2 *) (2 k\) sin2r 



., . ' E (k it T) V TI T- -7= 2I 



KI R\ k\ sin 2 2v *J cos 2 ? y] _ * s j n % 



and further, 



or, if preferred, 



t2 A2 ; o. 



(22) 



whereby the coefficients of corresponding terms in Ii and L have a common denominator. 



In this way we have determined all the quantities that we shall require to use. 



In the tables below we have given the force-components of the rectilinear portion of the current, and 

 the values of the quantities P , calculated for various values of and o ft. The special calculation 

 is only required for values of itt / between and 180, answering to values of-r between and 90. 



For 



T = m;c + t\j 



7f 



where ;;/ is a whole number, and t\ an arc <^ - , we have, for E and F, 



(see Legendre, 1. c., Vol. I, p. 14). For the third term we also have exactly the same relation, 



sin 2r _ sin 



yi k\ sin% yl ^?sir 



the only difference being that the value of the expression, for r = -^- is equal to zero. We therefore 

 have the relation, 



Finally we will also give the formula for the magnetic potential of the current. This can very simply 

 be deduced from the formula for the components of the magnetic force. 



