OiV THE HYPOTHESIS OF UNDULATIONS. 375 



mX 2 7r df, 



Suppose now that u, = cos — (at- x) -^~ , 



2 TT X rf<r 



m\ 2 TT ^ , df^ 



V, =-— cos— (at- z)-j- , 

 Stt \ dy 



m'\ Stt , d/, 



u., = cos — (at - z + c) -^ , 



m'\ S'jr , rf/, 



i> , = cos — (at - z + c) — . 



" 27r X ^ dy 



rru ^ ^'^ r '. ^ 'if' . df,\ X . 2-K , , , , . df.. 



1 hen ?<| + ?<_, = -— cos -- (a < - r) [m ~ + m cos c -^ sin — (at - z) m sin c ^-" 



2 7r X \ d.v dxl 2ir X d x ' 



■ . df., 



m sin c — 



2 TTP ti X 



tan — 



and if 



X rf/-, , (//•, 



»» -; — + m COS c -^— 



d X d X 



X f ,df,^ , df\ df. ,,df.. 



C7 or Ml + u, = — W~r¥ + ^wra COS c — . -^ + m -^, 



Stt I o;i' dx dx dar 



m sin c 

 ■27r0 dy 



so If tan 



I' 2,r , 



} COS — (at — . 



J X 



df. 



X d/, , d/,' 



m 1- m cos c — 



dy dy 



X I ,df^ , df\ df.. ,^dff\h 2 7r , 



K or U| + J), = — < m" — — „ + 2 m m cos f -^ . -^ k- m -^> cos — (o < - :^ + f^ ). 



Stt \ dy- dy dy dy'] X ' 



The two velocities U and V are not in this case in the same phase, and consequently the trans- 

 verse motion of a given particle, instead of being in a straight line, is in an ellipse or a circle. 

 The effects of the resolved parts of the velocities in the directions of the axes of x and y may be 

 assumed to be independent of each other, and the intensity of the compound ray will conse- 

 (|uently be as the sum of the squares of the maximum values of U and F; that is, as 



. [df^ dfS ^ ^, , /rf/. df, df dfA ,, idf? djT. 



\d.v dy^J \dx dx dy dy! \dx' dy' J 



which on account of the equation (23) is independent of cos c. Hence the intensity is the same 

 whatever be the difference of phase, and therefore the same as when the two polarized rays have 

 the same phase. This agrees with experience. 



Let us now proceed to consider the bifurcation of a polarized ray ; for instance, the ray whose 

 condensation is cti . Suppose it separated by any means into two rays whose condensations are t, 

 and T.,. We shall assume, as in the case of a ray of common light, that the sum of the conden- 

 sations at corresponding points of the divided rays is equal to the condensation at the corre- 

 sponding point of the original ray, and that the velocities at these points of the divided rays 

 are in directions at right angles to each other. We have then, by what has gone before, 



0-, = T,+ T, (21.), 



dri dT-, dr, dr. , , 



dx dx dy dy 



