Mr EARNSHAW, ON FLUID MOTION. 227 



The general inference from this reasoning appears to be this : that 

 when the integral of a differential equation contains constants (intro- 

 duced by integration), the values of which the proposed conditions of 

 the problem are not sufficient to determine, the singular solution* is 

 the proper integral for that particular problem. 



28. Upon these grounds I state the following general principle, 



A wave may, at any moment, be resolved into plane component waves, 

 each of ivhich is a tangent to the original wave. These components may 

 he supposed to he uniformly transmitted with the velocity a, and at any 

 time they may he co7npounded into a single wave hy taking that surj'ace 

 to which they are all simultaneously tangents. 



Hence if the form of a wave-surface at any one instant be known, 

 its form at any other time will be determinable from geometrical prin- 

 ciples. 



The thickness (or, as it is sometimes called, the breadth) of a wave 

 is never altered by transmission. 



Ex. 1. In a quiescent medium let us suppose one of its particles 

 to expand, pushing equally from its centre on every side the adjacent 

 particles. The effect of such a disturbance will be a sudden condensa- 

 tion in its neighbourhood, which we may divide into concentric spherical 

 wave-surfaces, for each one of which ^ is constant ; though from surface 

 to surface ^ may vary. Now resolve any one of these surfaces into 

 its components by drawing an infinite number of tangent planes to it ; 

 each one is transmitted with the velocity a parallel to its edge, and 

 thus at the end of any period they will be equidistant from the centre 

 of original disturbance, and be tangent planes to a spherical surface, 

 which is therefore the form of the wave at any moment. 



Ex. 2. In a quiescent medium let all the particles situated in a given 



* Or rather the solution determined by the usual method of finding the singular 

 solution, for as is well known such a one may not happen to be a singular solution 

 but a particular integral. 



Vol. VI. Paut II. G g 



