44 PROCEEDINGS OF SECTION A. 



To obtain a theoretical foundation for Gladstone's formula, we 

 have only to regard the question of the transmission of an 

 undulation in a discontinuous medium from a time point of view ; 

 if it is propagated more slowly through matter than through 

 ether, then it loses time in travelling throvigh a molecule — strewn 

 medium. The total loss of time by a wave will in a first approxi- 

 mation be proportional to the loss of time in a single molecule, to 

 the fraction of its surface which at any instant passes through 

 molecules, and to the length of the path travelled, and to the 

 number of molecules in unit volume. Let s be the length of the 

 path considered, I the mean distance through a molecule, a its 

 mean sectional area, m its mass and d the density of the medium, 

 so that dj'm is proportional to the number of molecules in unit 

 volume. If V is the velocity of liglit in free ether, and V in 

 the matter of the molecule, then the loss of time in a molecule is 

 l.jY — IJv, and we can write the whole loss of time as 



^ a sld / \ 1\,,. , , n.,-P • , 



k I ] , k being a constant. But it w, is the mean 



velocity of light in the medium then the loss of time is s/u^ - /su. 

 s s asld /^ 1 1\ 

 vi V m \Y vj 



(?i - 1 ) , = ^' ^« (N - 1 ) = constant 



wliere n is the index of the medium and N of the matter of the 

 molecule. This' is Gladstone's law. 



If 'inld is regarded as the domain of a molecule (usually called 

 the molecular volume), while la is the true volume of the molecule, 

 we see that if la is supposed identical with tnld, n becomes 

 identical with N, so that the value of k is unity, and the 

 above relation may be written in the symmetrical form 

 (?i-l)M = (N- 1)U. This is a first approximation only as we 

 have taken account only of delay produced and actual loss of 

 time in the molecule, there is a secondary cause of delay in the 

 wavering motion of the broken wave front. The amount of this 

 vanishes with the density and is proportional to the length of the 

 path, so that we may expr-ess it by s {hd + cd") where h and c are 

 constants which may depend on the arrangement of the atoms in 

 the molecule, and adding this to the right hand side of our 

 oi iginal equation we get 



S S Sald/1 Iv , /; 7 , V'\ 



= ( _ „\ + s (bd + cd-) 



Vi V «i ^ V v/ 



.: (n-lfl^la(N-l) + m{b + cd) 

 m {n - 1 )/d = constant + ed 



