460 Prof. Challis on the Dispersion of Light. 



combined with the fourth expressing the principle of constancy 

 of mass, a linear differential equation, having constant coefficients 

 and a for principal variable, was obtained antecedently to any 

 supposed case of motion, it follows that the mutual action of the 

 parts of the fluid is such as to be consistent with the coexistence 

 of small motions. 



Having thus indicated the processes of reasoning which con- 

 ducted to rectilinear axes of motion, to vibratory motion, and to 

 the coexistence of vibrations, prior to supposing any case of mo- 

 tion, and having elsewhere given in full the investigation of exact 

 analytical expressions for the vibrations parallel and transverse to 

 the rectilinear axis (see the Philosophical Magazine for May 

 1849), I have now to add a remark of essential importance in 

 the present inquiry. On passing from these antecedent general 

 inferences to the consideration of particular cases of disturbance 

 of the fluid, it is necessary to proceed on the principle that the 

 state of the fluid as to velocity and condensation is both initially 

 and subsequently consistent with the previously demonstrated 

 general characteristics. Thus, for instance, a series of plane 

 waves must be conceived to be composed of vibrations parallel and 

 transverse to rectilinear axes unlimited in number and all per- 

 pendicular to the plane fronts of the waves. This instance I 

 have especially considered in the article on Double Refraction 

 in the December Number, where reasons are also given for con- 

 cluding that in consequence of this composition of vibrations 

 lateral divergence is prevented, and the motion might be com- 

 prised within a cylindrical space of very small transverse section. 

 This result from the mathematical theory of undulations is sin- 

 gularly in accordance with the phenomenon of the transmission 

 of a thread or pencil of light, — a very remarkable fact, of which 

 I am not aware that any theoretical explanation had previously 

 been given. The result is also worthy of notice as presenting 

 an instance of small vibrations for which udx + vdy + wdz is 

 clearly not an exact differential. Further, it is to be observed 

 that for this composite motion, supposing it to be wholly parallel 

 to the axis of x, the dynamical equation becomes 



fo du 

 Ka 'dx + dt~"> 



the factor k? being introduced, as I have shown in the Philoso- 

 phical Magazine for November 1853, in passing from free to 

 constrained rectilinear vibrations. 



In the case for which the vibratory motion is central and is 

 supposed to be a function of the distance from the centre, ac- 

 count must still be taken of the same law of the composition of 

 vibrations. Since, as in the preceding case, the resulting motion 



