Energy in an Interference Field. 293 



§ 2. The form of the mean energy curves in che neighbourhood 

 of a point. 



We shall first investigate the curves " microscopically/' 

 examining them in a region whose dimensions are o£ the 

 order of magnitude of the wave length. On account of 



the great value of — all terms in the differential equation 



2-7T 



except cos — ■ (r 2 — r x ) may be regarded as constant for this 



investigation. Let r l = pi, r 2 = p 2 be the point in whose 

 neighbourhood the curves are to be studied. The differential 

 equation is then 



977- 



(pz 2 + Pi 2 cos 0)dr 2 + pip 2 (l + cos 0) cos — (r 2 — r\) (dr 2 — dr x ) 



= (pi 2 + Pi cos 0)dr u 



and its solution is 



\ 2tt 

 r 2 (p 2 2 + p^ cos 0) + p x p 2 (1 + cos 0) ~~ sin — (r 2 - r x ) 



— r i(pi 2 + p2 2 cos 0) + const. 



It maybe assumed without loss of generality that p — p x = m\, 

 where wis some integer. Then the equation of the curve 

 which passes through the point r 1 = p 1 , r 2 -=p 2 is 



itf + P {> cos 0) ^ 2 - (pf + p 2 2 cos 0) ^i 



= ~^ 2 (l + cos^)sin^^. 



>7T " ' X 



It will be convenient to introduce reotangnlar coordinates 



Fi-. 3. 



3 



a, y as new variables in this equation, such that (fig. 3) 



