Generality of a New Theorem. 141 



definite integrals, or rather of expressions each of which may 

 involve several definite integrals. 



However, to treat these matters adequately an author must 

 enter into mathematical details, and as I was writing a paper 

 which I hoped would be read by microscopists as well as 

 physicists, I thought it best to avoid every mathematical 

 detail and to exclude every abstract consideration, the dis- 

 cussion of which could be kept out of the paper without 

 impairing it for the special object I had in view, viz. the 

 explanation of microscopic vision. It was on this account 

 (that is, to make my paper intelligible to a wider circle) that 

 the theorem is in my memoir enunciated in the most limited 

 form that would serve the immediate purpose of interpreting 

 microscopic vision, and also that I had to content myself with 

 a bare statement of the relation between it and Fourier's 

 theorem. 



The proof, however, which is given in my paper completely 

 proves the theorem in its fuller form. It does not require to 

 be modified in any respect except that the term light must be 

 understood in the generalized sense of any kind of wave- 

 motion. 



Here, and in my correspondent's enunciation of the 

 theorem, the word motion must be understood in the ge- 

 neralized sense employed in my paper on " The Kinetic 

 Theory of Gas regarded as illustrating Nature/' which will 

 be found in the Phil. Mag. for October 1895, p. 336. It 

 therefore includes any event that may be propagated through 

 the medium in waves, whether motions proper, alternations 

 of electromagnetic or dynamic stresses, or any other, and 

 may therefore be represented mathematically by any function 

 of the coordinates and time. 



It may be useful to state here the limitations within which 

 the theorem is true. These are : — 



1. That the medium be uniform. It may, however, be 

 either monotropic or crystalline, see No. 4 below. 



2. That the motions (using the word motion in its gene- 

 ralized sense) be such that the simple geometrical super- 

 position at each point within the medium of the motions 

 reaching that point is legitimate : as is, for example, the case 

 with dynamical motions of small amplitude. 



3. That the whole energy be employed in the wave propa- 

 gation. This third condition excludes such a case as that of 

 sound in air (in which the theorem is, however, approxi- 

 mately true *); because in air the compressions produce heat, 



* There may be an escape of energy as referred to in No. 3, provided 

 it do not cause the values of the velocities a, b, &c, of No. 4 to include 



Phil. Mag. S. 5. Vol. 43. No. 261. Feb. 1897. M 



