View of the Undulatory Theory of Light. 109 



We should now have to substitute these values in the fun- 

 damental equations (12.), and thus obtain expressions involv- 



d^ 

 ing f -7^, &c., which would obviously extend to some length. 



But even without actually going through this process at length, 

 we shall easily perceive a principle of simplification arising 

 out of the form which we shall at once see certain parts of the 

 expressions must take, as follows : 



1st. In the forms (21.), all the terms involving f >j ? have in 

 their coefficients the square of the sine of a function of 8, and 

 these terms, when introduced as multipliers in (12.), are in the 

 first member uncombined with any other functions of the an- 

 gles a /3 y 8 ; and in the second members are combined with the 

 squares of the cosines of the angles ; that is, in every case these 

 terms are of even dimensions, 



2ndly. All the terms involving the differential coefficients 

 of f )j ^ have, in (21.), for coefficients the si7ie of a function of 

 8 ; and these in the multiplication also, in the first member, 

 stand uncombined with any other such function ; and in the 

 second, combined with the squares of the cosines ; that is, in 

 every case they are of odd dimensio?is. 



Also, it appears from the original construction and from 

 (18.) that the cosines of a /3 y S are all positive or all nega- 

 tive together. 



Now, in taking the sum of a number of terms (indicated 

 by the symbol S), it is evident in the former of the above two 

 cases that all those terms will be positive whatever be the signs 

 of the cosines. In the second case, for the same reason, the 

 terms will be positive or negative according to the change of 

 sign in the cosines. 



If, then, we suppose in such a sum, half the terms corre- 

 sponding to positive, and half to negative values of the co- 

 sines, we shall find that the coefficients of all the terms in the 

 second case will disappear, whilst in the first case they will re- 

 main. The whole expression will thus be reduced to the 

 terms involving >] ? only. 



This last supposition is precisely that of a physical condi- 

 tion which we shall have no difficulty in allowing, viz. that in 

 the state of equilibrium the masses of the molecules tn 7n' m"y 

 &c., are two and two equal, and distributed symmetrically on 

 each side of the molecule m on straight lines passing through 

 m. This obviously gives the cosines for half the molecules 

 positive, and for half negative. 



In such a case then we shall have the general equations of 

 motion reduced to a considerably simplified form ; or, for ab- 



