42 Mr. E. Gold : Relation between Periodic Variations of 



Then the equations for B/, B 2 are 



tan $ ^ - B/ tan <j>(y + t ± ) — /3i/rB 2 tan <£ + 2?iE sin 3 <£ 



— t y- F cosec (f> (4 sin 2 <£ + 5) = 0, 

 and 



tan ^L_?— B a tan £(y + g + /3^B/tan 6+ — ^ sin 3 <£ = 0. 



Put B^LTi^, B 2 = MT 1 6 Z . 



Then 



&L -M/^ + ^*- z [>E sin 3 <£- If cosec <£(4 sin 2 <£ + 5)]=0, 

 ^ + L/3^ + ?^/3^ *r« sin 2 <j> cos = 0, 



whence 

 where 



|^^Ul c ^ 2 r^-P c /3r z =i 



^ E&cos<f> ft . K . 4??E . 9 



P = — r (4t sm 2 + 5) -f -™-sm 2 <£ cos 6. 



2m^ siir</> r T 2 7- r 



If ^T x is taken to be independent of e, the solution of the 

 equations can be put into a form that admits of calculation. 

 In fact 



M=Qcos* + Ksin* + ^ 



L = -Qsint + Rco S t+g^[l- ( ^^y + ^ ) + ...] 



2wE/c . 9 . 

 e~ z snr a> cos 6. 



Here Q and R are constants to be determined from the con- 

 dition L = M=0 when z=0. 



Since L = at z—0, it follows that ^— is always negative 



near the surface, and therefore M is always negative near 

 the surface. Consequently B 2 and C 2 are always negative 

 near the surface. 



But -—-vanishes and changes sign when 

 O z 



2nE sin 3 <£ = 5- F cosec <f> (4 sin 2 <£ + 5) , 

 which gives $=72° nearly. 



