140 Dr. G. J. Stoney on the 



In this enunciation thejwords " of given period " are unneces- 

 sary and may be omitted. 



I had this generalization in view when writing the 

 paper, and it was it that led me to state on p. 335 that my 

 theorem is in ultimate analysis an extension of Fourier's 

 theorem. In fact to show this, it only needs to enunciate 

 the new theorem in its symbolical form, when it becomes 



g(«, y ^o=SAri°(ftr te+w \ +wg "' t +«) 



where the A's are directed quantities ; and where, for each 

 periodic time \/v, there will in general be three values of A 

 and a in each direction Imn. These belong to the normal and 

 two transverse waves, or to whatever correspond to them. 

 This equation may be written 



F (*, y, z, t) = 2 A sin (2tt V -^- + a\ , 



where p is the perpendicular distance from the origin to the 

 plane whose director cosines are Imn and which passes 

 through the point xyz. 



The analogy of the theorem when in this form withjjFourier's 

 expansion for a string vibrating in one plane, viz. 



F (*, = % [A„ s in(2n W ^=^+ «„) + B. sin (Zm: 2±* + £,)] , 



is at once apparent, and becomes still clearer when we 

 remember that the two terms in the Fourier's expansion for 

 each value of n correspond to the two opposite directions 

 of a line, the only possible directions in dealing with a line. 

 These are of course only particular cases of the directions 

 designated by the director cosines Imn of the new theorem, so 

 that the new theorem may legitimately be described as an 

 extension of Fourier's theorem. 



It may further be remarked that the new theorem has 

 somewhat similar relations with expansions by Spherical 

 Harmonics, by Bessel's Functions, et hoc genus omne. This 

 is to be expected ; for just as Fourier's Theorem enables us 

 to analyse the displacements of a line, and as Spherical Har- 

 monics &c. deal with displacements of certain surfaces, so 

 does the new theorem deal in a similar manner with displace- 

 ments or motions pervading a space. 



In one respect the new theorem is less perfect than its pre- 

 decessors, Fourier's Theorem, &c, since as yet the constants 

 of the new theorem have not been evaluated in the form of 



