330 The Rev. Professor O'Brien on the 



sical points placed at equal distances from each other. Then 

 each of the slices will oscillate parallel to the plane ofxi/, the 

 velocity being the same at every point of the same slice, and, 

 according to what has been above stated, each slice will expe- 

 rience a resistance to its motion depending in magnitude and 

 direction upon the velocity of the slice, the differential coeffi- 

 cients of the velocity with respect to the time, and the curva- 

 ture of the path described by any point of the slice (observing 

 that every point of the slice describes the same path, or, in 

 other words, that the motion of the slice is simply one of trans- 

 lation). On account of the extreme smallness of the intervals 

 between the resisting particles compared with the length of the 

 wave, the internal agitation of the aether composing any slice 

 will be inconsiderable compared with the general motion of 

 translation of the slice; something like the ripple on the sur- 

 face of a great wave of water. 



The forces which act upon any slice will arise from three 

 causes: — 1st, the unequal displacements of the slices from their 

 equilibrium position parallel to the plane of .c ?/ ; 2nd, the ge- 

 neral resistance to the motion of translation of the slice caused 

 by the resisting particles ; 3rd, the disarrangement of the aether 

 due to the internal agitation. The forces arising from the 

 third cause we shall neglect, on account of the smallness of 

 the internal agitation, and because these forces will be of the 

 nature of internal forces, and therefore produce little or no 

 effect on the motion of translation of any slice. 



9. Let s be the distance of any particular slice from the 

 plane of xy, and ^>j its displacements parallel to the axes of x 

 and j/. Then the accelerating forces parallel to the axes of x 

 and J/ due to the first of the causes above mentioned will be 

 (as is well known) 



d^^ d^ 



^dz^' ^dz^' 



A being a constant depending upon what may be called the 



lateral or transversal elasticity of the aether. 



Let the unknown resistance on the slice be resolved into 



two forces T and N, the former acting parallel to the tangent 



of the curve which any point of the slice describes, tending to 



check the tangential motion of the slice, as we may call it; and 



the latter acting parallel to the normal tending to check the 



motion of deflexion towards the centre of curvature. Then, 



, . , 1 d^ J 1 dri / , , /d^Y /dy}Y\ 



observmg that r^ and r- ( where v^=[-r:)+[-r) ) 



^ V dt V dt\ \dtJ \dt/ ) 



are the cosine and sine of the angle which the tangent makes 

 with the axis of a;, we have the following equations of motion : 



