170 The Rev. J. Challis on the different Refrangibility 



out affecting each other. Hence there will be one point at 

 least, at which the velocities arising from the two propagations 

 will be equal and opposite, and consequently the resulting ve- 

 locity be nothing. At this point suppose an indefinitely thin 

 rigid partition to be placed transverse to the axis of the tube, 

 so as to divide the fluid into two separate columns. By this 

 supposition the state of the motion will in no respect be al- 

 tei'ed. But the column on one side of the partition cannot 

 affect that on the other; therefore if the one be removed, the 

 motion of the other will remain unchanged. In this case the 

 partition serves as an obstacle against which the undulations 

 are reflected; and we are thus taught that the reflected waves 

 are exactly like the incident, and are in fact a continuation 

 of the incident, but diverted into an opposite direction of pro- 

 pagation. Let the law r of the velocity v and condensation 5 of 

 the incident undulations be given by the equations, 



v = — a s = — m sin — (x + a t), 



in which x is measured from the plane of reflection, in the 

 direction of the propagation of the reflected waves, and t is 

 dated from an instant at which condensation commences at 

 this plane. By changing the sign of a, we obtain the equa- 

 tions, v' = a s' = — m sin — [x — a t), 



applicable to the reflected waves. At any point distant by x 

 from the reflecting plane, the velocity at a time t will be 



v -f v' =s — 2 m sin cos , and the condensation s 4- s' 



= cos sin . Hence if <r = the condensation at 



a X X 



the origin of x. a<r = 2 m sin l^— t 



I will just remark, that the foregoing method of solving the 

 problem of reflection may be readily extended to motion in 

 space of three dimensions, by conceiving undulations to be 

 propagated under circumstances exactly alike from two sepa- 

 rate centres, and the fluid to be divided by an indefinitely thin 

 rigid partition, bisecting the straight line joining the centres 

 at right angles. The principle of the division qfjluids, as it 

 may be called, which is here made use of, is legitimately em- 

 ployed, because the possibility of such separation of the parts 

 of fluids, without affecting the state of motion or rest, forms a 

 characteristic property by which they are distinguished from 

 solids, and might be made the foundation of the mathematical 

 treatment of them ; since the equality of pressure in all direc- 

 tions 



