AS EXPLAINED BY THE HYPOTHESIS OF FINITE INTERVALS. 175 



We have already deduced from tlie expression for the velocity a 

 reason for its uniformity in vacuo. The necessary condition was, that 

 he distance between the particles in vacuo should be much smaller 

 than ,n refractmg media. We will now recur to this point, and con- 

 sider the subject in rather a different light. However it may arise, 

 this is certain, that if 1 be expanded in a series of the form 



q 



the quantity q vanishes with respect to p in vacuo, whilst it does not 

 in refracting media. On no hypothesis that I can conceive, of the 

 action of forces producing undulations, should we expand our functions 

 m a series descending by powers of A, and also by those of the dist 

 ance between two particles, for we must approximate by considering, 

 one quantity smaU compared with another; nor can I consider a series 

 ascending by powers of the ratio of X to the distance between two 

 particles, as it would involve an absurdity. There seems, then, every 

 reason to suppose that the form at which we have arrived is the cor 

 rect one. I can indeed conceive it possible, and not at all improbable 

 that the particles of which the substance is composed should influence 

 the motions of the particles of ather. But should they do so, the form 

 of our functions would not be affected; and the only difference would 

 be, that ^, would equal A^ \^{r) + E^ S^,^ Sp\ a quantity independent 

 of the distance between two particles of aether, and varyino- only by 

 the pecuhar constitution of the material particles composino- the me 

 dmm. This, then, cannot in the slightest degree affect our reasoning. 



Having, then, sufficient ground for the adoption of the results in 

 their present form, we will proceed to re-examine the expansion from 

 which ]}, q and I were derived. The general form is 



^,=A^{cp{r) +r^^^'-'|sin.::^_ 

 Now in expanding the sine of an arc it is requisite that the arc itself 



