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XXXVIII. Discussion of a New Theorem in Wave Propagation. 

 By G. Johnstone Stoney, M.A., D.Sc, F.R.S.* 



IN the first of three papers on Microscopic Vision in the 

 Philosophical Magazine of last year, on p. 335 of the 

 October number, the theorem is enunciated that any disturb- 

 ance in the luminiferous aether may be resolved into undula- 

 tions of plane waves, each wave being of unlimited extent and 

 uniform throughout its extent. 



The theorem is of course not limited to disturbances of the 

 special kinds that can exist in the luminiferous sether, but 

 applies generally to any disturbance within a given space, or 

 to any system of displacements that may prevail throughout 

 it. (See the Philosophical Magazine for February 1897, 

 p. 139.) 



The symbolical expression of this theorem is that 



where dco is the element of solid angle round the direction Imn. 

 If written in polar coordinates the theorem takes the some- 

 what simpler form 



F(.r, y, z, t) = $2 [ M sin (*2tt ^^ + «Y] . sin Q dO d<j>. 



In this enunciation of the theorem r stands for Ix + my + nz, 

 or its equivalent x cos + y sin cos (/> + z sin 6 sin <f> ; in it 

 also vt is written instead of the wave-length X, the suc- 

 cessive values of t being the periods of the successive terms. 

 M the transversal of one of the terms, a. its phase, and v its 

 velocity J, are all functions of 0, (/>, and r. The transversal M 

 is a directed quantity and of such a kind that it may be 

 resolved into three principal positions, which will usually be 

 the position 6<f) and two others at right angles to it and to one 

 another. Hence the aggregate of all terms like that within 



* Communicated by the Author. 



t This way of writing- the theorem is to be preferred to that given on 

 p. 140 of this volume of the Philosophical Magazine, because by writing 

 Mg?oo instead of A it keeps clearly before our minds the circumstance that 

 when treating of a resolution into plane waves the elements of the sum- 

 mation are not individual undulations of plane waves but sheafs of such 

 undulations (see the November number of the Philosophical Magazine, 

 p. 436). So, in plane geometry, the element of area is not a line y but a 

 strip ydx. 



% In monotropic media where the wave-surface is a sphere, v for each 

 position of transversal becomes a function of r only, 



Phil. Mag. 8. 5. Vol. 43. No. 263. April 1897. Y 



