an Electric Source, and Line Spectra. 789 



zeros of radiation, the wave-lengths X lt \ 2 , &c. being almost 

 imperceptibly greater than v l5 v 2 , &c. Thus, the maxima of J 

 will be attained for 



U = Ui — 8{U, 



where 8{U are very small fractions. 



By (23), Second Paper, J attains its maxima when u 

 becomes, very nearly, a root of the equation 



cos u , K __ n 



i. e. when 



, N u 2 COS u 

 9W= £-• 



Here we can, with sufficient accuracy, put on the right hand 

 u equal to the successive critical values u lt u 2 , &c. 



Thus, the relative emissivity of the source when emitting 

 waves (X = X { , i==l, 2, 3, &c), for which J attains a maxi- 

 mum, will be given by 



47T Ui G COS 2 Ui /nn ^ 



ei = W^ { ' *= l > *>*>**-> • • • < 33 ) 



where, by (9), and because sin 1^/^= cos Ui, 



G,=G(^)=t£ i ~^!ifi+2Si(2w.). . . . (34) 



Introducing here the known values of v^, u 2 , &c, and using 

 the tables of Si already quoted, w r e find, for the (common) 

 logarithms of Gx to G 6 , 



0-83000, 1-01337, 1-13163, 1-22583, 1-30359, 1-36721. (35) 



The values of 8iU, determining the distance of the maxima 

 from the corresponding zeros of radiation, will be very nearly 

 given by 



/ X \ / d 9\* Ui 2 CO<Ui 



9{l(i -8 1 n)=-^8 i u = K -. 



Now, y = sin ulu — cos u y 



dg . 1 f . /dg\ 



-f- = sin u q{u) ; ( ~ ) =sinw.-. 



du u* K ' ' \du U 



Therefore, 



B.u= j^, •=l,2,3,&c, . . . (36) 



These approximate formula?, (33) and (36), are more than 



