622 Mr. S.N. Basil on ilie Deduction of Bydberg's Law 

 Ihe integration. 

 From (1) we have obviously 



Uijl 



of the two remaining expressions, (3) can be integrated 

 most easily : in fact, 



f\ 



nJi = 1 - V — W?//r 2 + 2me 2 r — c 2 dr 

 J r 



gives after integration 



nji = 2tt 



/.(*)-^]- 



The second integral can be written as 



n 2 li r C i~ — — ^ — — « — ~ — =f — —* ^ dx 



2 



= I = fVfe-^ 2 )- r, 2 x*-2me\ux{l-x*) f 



"by putting cos 6 = x. 



The right-hand side is to be integrated throughout the 



region, when the cubic remains positive. It cannot be 



integrated in finite terms ; an approximation suitable for 



our purpose can, however, be made, assuming 2mth to 



be small compared with (c 2 — c^) = A. To see what this 



means we are to remember that c 2 — c 2 is of the dimension 



of h 2 ; so that 2meh must be small compared with A 2 , or 



h 2 

 L must be small compared with — . Now, if a, — /3, and y 



me 



are taken as the three roots of the cubic, the cubic can 



be written down as 



D(7-*)(«-#X* + £)> 



where 7 is the greatest of the positive roots and D = 2m<>L. 

 The limits of the integral are obviously ol and —/3, 



J-/3 - 1 - — A 



Hence 



or 



B_I _ L f tt -.T<fa 



3D ~ 2 J-, V D[( 7 - t i-)(a- t ,)( t f + ^] 1/2 ' 



3J 1 r f « (y-nfPdx 



f a <*# 1 



7 J-iB (7-.t') 1/2 («- *0 1/2 (*+/3) 1/2 J 



