323 Prof. Challis on a Mathematical Theory 



above referred to, the caloric repulsion of the individual atoms 

 of substances is ascribed to the dynamical action of this class of 

 waves. 



When, however, the excursions of the vibrating particles of the 

 fluid are large compared to the diameter of the sphere, it is con- 

 ceivable that the condensations and rai'efactions on the second 

 half of its surface may, by the propagation of the momentum, 

 become equal to, and even exceed, those on the first, and that 

 the sphere may receive a permanent motion of translation towards 

 the origin from which the waves are propagated. 



It would seem, therefore, that the general solution of the pro- 

 blem above enunciated would give the means of forming at once 

 a theory of forces both attractive and repulsive. It is not, how- 

 ever, my intention at present to attempt to solve the problem 

 generally, my object being rather to inquire under what circum- 

 stances and in what manner a permanent motion of translation 

 of the sphere may take place in the direction contrary to that of 

 the propagation of the waves. With this view I propose to dis- 

 cuss separately several hydi'odynamical questions, the answers to 

 which will perhaps give a more distinct idea of the dynamical 

 action by which such a motion may be produced, than could be 

 obtained by treating the subject more generally. The solution 

 thus arrived at will, however, be subsequently shown to be con- 

 sistent with the general hydrodynamical equations. 



It is right to premise that in the following investigations I 

 take for granted, as already proved, the laiv of axes of rectilinear 

 propagation of vibratory motion which I have recently applied in 

 the explanation of certain phsenomena of light. In consequence 

 of this law, any vibratory impulse communicated to the fluid is 

 rectilinearly propagated till its course is diverted by the reaction 

 of some solid obstacle. Such rectilinear propagation is entirely 

 compatible with curvilinear motion of a given particle. In the 

 case of a stream of uniform density, the motion may be consi- 

 dered to be that to which vibratory motion approximates in pro- 

 portion as \, the breadth of the waves, is increased, and therefore 

 to be subject to the same law of rectilinear transmission. 



Pkoblem I. To find the relation between the condensation (cr) 

 and velocity (V) in a series of plane-waves propagated in a given 

 direction, terms containing the square of the velocity being in- 

 cluded. 



According to principles which I have already on several occa- 

 sions explained, the plane-waves must be regarded as compounded 

 of an unlimited number of ray-undulations, in the same phase of 

 vibration, and having their axes all parallel to the direction of 

 the propagation of the waves. The cfi'ect of this composition, 

 which destroys the motions transverse to the axes, is taken into 



