404 Mr. O'BRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 



As we have just explained, a may be considered to be a continuous 

 function of x, y, »', t ; and a is its value when x' = x, y = y, z = % ; hence, 

 by Taylor's Theorem, we may suppose a - a or $a expanded in the form 



da . da . da s «* 2 a &«" d" 2 a II d s a Sy" 



d-x Sx + d^^ + d-J z + d^^- + dx-r y ^^ + dr f + &c - &c -; 



if we substitute this value of $a in equation 2?, -7- -=- -=- -=-^ &c. 



^ «# ay as aV 



may all be brought outside 2, and in the same manner, if we sub- 

 stitute similar expressions for S/3 and iy, the partial differential co- 

 efficients of (& and 7 may be brought outside 2: the result of these 



d* a 

 substitutions will evidently be a linear equation between -7— , and the 



successive partial differential coefficients of a, /3, 7 with respect to 

 x, y, %, multiplied by such quantities as 



2mf(r)§x, *Lmf(r)Sx\ Zmf(r)Sx§y, &c. &c. : 



these quantities will evidently be, in general, different for different 

 particles ; that is, they will be functions of x, y, % ; hence the equation 

 (2?) will, in general, become a linear differential equation with variable 

 coefficients. 



We cannot determine what functions of x, y, % these coefficients 

 are, since to do so we ought to know the law of force of one particle 

 on another, and the manner in which the particles are arranged when 

 in a state of equilibrium, neither of which things we know ; hence it 

 appears impossible to make use of the equation (2?) unless we employ 

 some hypothesis to simplify it. 



§ 9- The hypothesis which naturally suggests itself is, that of a 

 symmetrical arrangement of the particles when in a state of equilibrium. 

 But there is a difficulty here, arising from the influence that must be 

 exerted by the material particles on the arrangement of the etherial 

 particles; for supposing the material particles symmetrically arranged, 

 it is evident that if there be a number of etherial particles surrounding 

 each material particle, the arrangement of the former cannot be symme- 

 trical ; for they will be disturbed from their positions of symmetry by the 



