274 Dr. G. J. Stoney on a New Theorem 



brackets in the formula, in any one direction 6$, is in effect 

 the sum of three separate series, corresponding to the three 

 principal positions of the transversals of waves travelling in 

 that direction. There are accordingly three times as many 

 terms in the summation as there are values of r. If the 

 original disturbance F(x,y, z, t) is periodic, each of the three 

 series may be further simplified by supposing t to take in 

 succession all the values of T/?i as in Fourier's theorem, T 

 being the period of the disturbance and n an integer. But it 

 seldom happens that F(x,y,z, t) is periodic. If it be not 

 periodic, the simplest conception to entertain is that M and a 

 vary either abruptly from time to time or continuously ; but 

 practically a much better treatment, though not the simplest, 

 is to regard the values of t as indefinitely close to one another*, 

 and M and a as not varying with the time. This is equiva- 

 lent to regarding T as infinite. 



Every theorem of this kind can be investigated in various 

 ways; and these will famish proofs, some of them symbolical, 

 others geometrical. The symbolical type of proof is chiefly 

 of use if it can be made to furnish expressions for the M's, 

 the a } 8j and the v's in each direction Imn. It may be hoped 

 that this will soon be accomplished. The chief value of the 

 geometrical form of proof is that it gives us a more continuous 

 view of what is going on in nature, inasmuch as the steps of 

 the geometrical proof of a physical problem keep throughout 

 their whole progress in close proximity to what actually takes 

 place, whereas a symbolical proof is in contact with nature 

 only at its commencement and at its close. The intermediate 

 steps are in general mere paper work. Each method accord- 

 ingly has its special advantage, and it is desirable that both 

 shall be studied. Two examples of geometrical proofs of the 

 theorem have been given in the October number of the Philo- 

 sophical Magazine, one on p. 335 and one in the footnote on 

 the following page, but neither of these is the most direct of 

 that kind of proof. The geometrical proof seems to reach its 

 simplest and most direct form, and therefore the form which 

 gives to us the fullest insight, when presented as follows : — 



Alternative proof of the theorem that Any disturbance 

 within a given space may be resolved into undulations of uniform 

 plane loaves. 



* By regarding matters in this way it can be seen that, where lines in 

 a spectrum arise (as they always do) from a jumble of molecular events, 

 they must each have some physical width in the spectrum : even in the 

 case where the molecular events taken separately emit waves of strictly 

 definite periodic times only, which if it were not for the breaks and inter- 

 vals between the events could only furnish lines in the spectrum devoid 

 of physical width. 



