on the Hypothesis of Undulations, 469 



to r— an arbitrary value; and it is readily seen that the two 

 latter integrals are each equal to half the other. 



Having recently found that the resolution of a polarized ray 

 into two parts admits of a much more simple treatment than 

 that referred to in art. 42, I propose to introduce here the dis- 

 cussion of that problem. There are only two conditions which 

 the resolved parts of a polarized ray are required to satisfy, viz. 

 that the sum of their condensations at corresponding points be 

 equal to the condensation at the corresponding point of the 

 integral ray, and that the velocities at corresponding points be 

 the parts, resolved in directions parallel and perpendicular to the 

 new plane of polarization, of the velocity in the integral ray at 

 the same corresponding point. Let that plane make the angle 

 6 with the axis of x, and let s, a l} <r 2 be the condensations at 

 any corresponding points of the original and resolved rays, and 

 f> fv f<2 De the factors for the same points, the values of which 

 must satisfy the equation (10) in art. 26. Then, if we assume 

 that cr l =5COs 2 ^ and <7 2 =ssm 2 #, we have a l + (T <2 =s, and the 

 first condition is satisfied. Also it will appear from the follow- 

 ing considerations that the other condition is satisfied by the 

 same suppositions. Let S, Sj, 2 2 be the condensations at the 

 points of the axes cut by the transverse planes in which the con- 

 densations are s, <r v cr^. Then, from what is shown in arts. 39 

 and 40, 



f^i = <T ] =s cos 2 =/S cos 2 9, 



Hence 



also 



/ 2 X 2 = c 2 = s sin 2 =/S sin 2 6. 

 & ax T ax b ax T ax 



The left-hand sides of the last two equations are the parts of the 



velocities in the bifurcated rays resolved parallel to the axis of x. 



The other sides of the equations are resolutions of the velocity 

 if 



(f>-~f- of the original ray (which is by supposition wholly parallel 



to the axis of x) into parts parallel and perpendicular to the new 

 plane of polarization, which parts arc then resolved in the direc- 

 tion of the axis of x. Hence the velocities parallel and perpen- 

 dicular to that plane, which are the velocities of the bifurcated 



rays, are respectively equal to </>y-cos# and <£y-sin#, that is, 

 the parts of the original velocity resolved in those two directions. 



