a Theorem in Wave-motion. 101 



planes of the waves to which they belong. Each of these 

 trains of plane waves the luminiferous aether can propagate 

 forward unaltered, whether it exist alone in the aether or 

 accompanied by other motions. Finally, this analysis is of 

 great use for investigating numberless optical problems. 



Mr. Preston's analysis decomposes each transversal into 

 its £, t], and £ components. Of these he forms three groups. 

 He groups all the £ components together, all the rj compo- 

 nents together, and all the £ components together ; and by 

 his theorem he decomposes each of these groups into trains 

 of non-natural waves with lengths ranging from infinity 

 down to cypher, and with transversals which, in all but three 

 directions, stand obliquely to the front of the wave. Trains 

 of waves of this kind, if isolated, cannot be propagated by 

 the sether. The instant one of them is left to itself it breaks 

 up, and what is thenceforward propagated is something quite 

 different. Finally, the analysis is of no known use in physics. 

 From all this it may be judged how much what Mr. Preston 

 says about this case, at the bottom of p. 259 and top of p. 260, 

 needs correction. 



The useful theorem — that which supplies us with a resolu- 

 tion into uniform plane waves which the medium is compe- 

 tent to propagate — has been established by the geometrical 

 method of proof which we owe to MacCullagh, and which is 

 based on the Principle of Reversal. No analytical proof 

 has yet been discovered, but the difficulties in the way of 

 such a proof seem such as ought not to be insurmountable. 

 We have seen, on p. 273 of the April number of this 

 Magazine, that the symbolical expression of the results 

 arrived at by the geometrical proof is 



FO, y, z, t) = (T t [M sin f 2vr ^^ +*)!. sin 6 dd d<f>, 



where r = x cos 6 +y sin 6 cos <£ + z sin 6 sin </>, and in which 

 the M's, the transversals, are directed quantities. In this 

 expression the vector components of the M's may be carried 

 outside the sign 2, but no farther. Accordingly, the double 

 integral is of a new kind, in which the summation is a geo- 

 metrical summation, and not an algebraic summation as it is 

 in ordinary integrals. This presumably is the source of the 

 difficulty. It seems, however, to be a state of things which 

 may be dealt with, if we can succeed in finding a dynamical 

 problem in which the M's shall be forces, but which shall 

 lead to the above symbolical expression in all other respects . 

 and there is no apparent reason for regarding it as hopeless 



