312 Mr. Challis on the Theory of the 





+ &c. to as many terms as we please. 



For each of these terms satisfies separately the differential equa- 

 tion of the motion, and gives the motion resulting from the 

 disturbance to which it belongs. Also the terms taken collec- 

 tively satisfy the same equation, and give the motions consequent 

 upon all the disturbances acting simultaneously. We have thus 

 a proof of the principle of the co-existence of small vibrations, 

 and infer from it, that the motion which a particle has at any 

 instant in consequence of several disturbances, is the resultant of 

 the several motions it would receive, if each disturbance acted 

 alone exactly as it does contemporaneously with the others. 



18. Conceive two points to be disturbed under circum- 

 stances precisely similar, so as to be the origins of two series 

 of waves exactly alike. Then, by the Proposition just proved, 

 the particles in a plane bisecting at right angles the line joining 

 these two points, will move entirely in this plane, and will 

 have no motion perpendicular to it. If, therefore, in the place 

 of the imaginary plane be substituted an indefinitely thin rigid 

 partition, the motions of the particles will not be altered. But 

 plainly the disturbance on one side of the partition cannot affect 

 the particles on the other. Hence the effect of such disturbance 

 is supplied by reflection at the partition. The angle of incidence 

 is equal to the angle of reflection, and the reflected waves are 

 exactly what the incident would have been, if they had gone. 

 on without interruption. We are thus informed, in what maimer 

 a .series of waves acts dynamically on any obstacle whatever that 



