472 The Astronomer Royal on the Equations 



the same mechanical equations referred to the same directions 

 in absolute space must apply to all these displacements. 



In ordinary crystals or fluids possessing the property of 

 causing rotation of the plane of polarization in the same direc- 

 tion as referred to the eye of the observer, whether the ray 

 be incident on one side or on the other, mechanical equations 

 are to be sought which will produce the result, that in both 

 cases the velocity of Ray No. I. is greater than that of Ray 

 No. II. (or vice versa) ; so that if U\ is greater than v" v v 9 

 will also be greater than w" 2 . But in the glass affected by 

 magnetism, if in the first experiment the velocity of Ray 

 No. I. is greater than that of Ray No. II., then in the second 

 experiment the velocity of Ray No. I. must be less than that 

 of Ray No. II.; or if z/j is greater than v" lt z/ 2 must be less 

 than ir a . 



Now the equation which is deduced from every mechanical 

 supposition that accounts for the propagation of undulations, 

 is of the form 



d*Y _ d*Y 

 dt* ' dx 



,2' 



& Z _ a d * Z 



dt* ~ dx*' 



And it seems probable that these equations, with the addition 

 to each of a small term, may explain the difference of veloci- 

 ties of the Rays No. I. and No. II. 



It was pointed out by Prof. MacCullagh, that the equations 



d*Y _ d*Y n d?Z 



dt* ~ dx* + 'da?' 

 d*Z _ <PZ_ B»ff 



dt* ' dx* dx 3 



would explain this difference. I may remark here, that in 

 the last term of the second side of each equation, any differ- 

 ential coefficient of an odd order would have sufficed to ex- 

 plain the general fact of difference of velocity; but the third 

 order was adopted by Prof. MacCullagh in order to reconcile 

 the expression for difference of velocity in differently-coloured 

 rays with the fact established by experiment. 



It is however necessary to inquire whether, if this assump- 

 tion makes v t l greater than v" v it will make fc/ 2 grater than 

 t/' 2 . For this purpose we must convert the various expres- 

 sions into expressions referred to the same co-ordinates. 



Let x x =x, jf 4 =5— x; y x -y, y«=- y\ 



in the first experiment let 



