SlO Mr. Challis on the Theon/ of the 



vary inversely as the distances. Let r be. equal to the internal 

 radius of a spherical shell of fluid, forming a part of the wave, 

 S = its thickness, supposed very small, and l + s its density : then 

 its mass = 4Trr-S (l + s) = 47rr-S very nearly; and its velocity 



varies as - . Hence its vis viva is the same at whatever distance 

 r 



it be from the centre, if ^ be the same. Hence also the vis viva 



of the whole wave is constant during its propagation, because 



its breadth is constant. The same thing is easily proved of waves 



propagated in space of two dimensions. 



16. Although the waves propagated from a disturbance made 



at a single point are always bounded by a spherical surface, 



because the velocity of propagation is always equal to a, the 



velocities and condensations of the particles are not the same 



in all directions, unless the disturbance be similarly related to 



all the parts of the surrounding space, as in the instance adduced 



in the preceding Art. The motion will be given in general by 



V = as = ~ F{r— at), 



and the form of F, when applied to the same wave, will be always 

 the .same in the same direction, but will be different in ditFerent 

 directions, according to a law depending entirely on the nature 

 of the disturbance. Conceive a pyramid tlie vertical angle of 

 which, formed by four equal plane surfaces, is indefinitely small, 

 to be placed with its vertex at the point of disturbance. The 

 particles within the pyramid will perform their motions as they 

 would if it were removed, because they move in lines directed to 

 its vertex, and the .sides of the pyramid, supposed indefinitely thin, 

 supply the pressure which upon its removal would be exerted 

 by the contiguous fluid. Now the motion within the pyramid 

 is strictly such, that the velocities and condensations at all equal 

 distances from the vertex are equal, and vary inversely as the dis- 



