406 Mr. O'BRIEN, ON THE PROPAGATION OF LUMINIOUS WAVES 



§ 10. I shall now make use of the hypothesis of symmetrical arrangement 

 to simplify the equation (B), which hypothesis, as I have shewn, necessaiily 

 implies that there are not so many, or at most, as many particles of 

 ether as there are of matter, or that the etherial particles are formed 

 into globules by the repulsion of the material particles. But I make use 

 of this hypothesis merely for the present in order to make myself more 

 readily understood. I shall hereafter prove that the results obtained 

 by means of this hypothesis are equally true when the arrangement of 

 the etherial particles is unsymmetrical, in consequence of the influence 

 exerted on them by the material particles. 



$ 11. To simplify the partial differential equations by means of the 

 hypothesis of the symmetrical arrangement of the etherial and material 

 particles. 



If the particles be all arranged symmetrically, it is evident that we 

 may so assume the axes of co-ordinates that the arrangement of the 

 particles shall be symmetrical with respect to them. This being the case, 

 it is evident that, if F(r) be any function of r, 2.m\F(r)$x t '$y''$z'\ is 

 zero unless each of the indices p q and s be even, and if each of these 

 be even, then this sum is the same for every etherial particle, i.e. it is a 

 constant. Moreover, p q s may evidently be interchanged without 

 altering the value of this sum. 



Hence if we put 



mM = Vm{f(r)$a?\, mN - S.mi-f'(r)SafSf\, mP = 2»t ji/'(r)^4 



the partial differential equation becomes (omitting at present .the part 

 under the sign 2 ) 



1^ d?a _ M id^a d*a (Pa\ P d*a. 



m ~dt i ~ ~2 \da? + df + dz* I + 2 da?' 



2 \dy- dtf dxdy dxd%) ' 

 + differential coefficients of the 4 th and higher orders. 



Now there is a very simple relation between P and N: for put 

 for a moment $x = u cos 6, $y = u sin 6, then (in virtue of the sym- 

 metry of the arrangement) for each value of u and r, admits of a 

 set of equidifferent values whose sum is 2tt; therefore 



