in the Problem of Sound, 403 



of that generality to secure the conditions that the general 

 value of X shall not contain at + x, and that when a? is =0 the 

 value of X shall have strictly the form c./{at) ; then the value 

 of X to the second order will be 



X=c./(^/~^)+^.;r.{A+/V-^)V}, 



where A is an arbitrary constant which may be used to remove 

 the non-periodical part oif'{at—xjf. Thus if 



f{at—x)= sin n{at—x\ 

 then 



f\at—x)]^=zn^. cos^ n{at—x)-= o" V + ^^^ 2n{at—x)h 



and here it will be proper to take A= — — ; and then the 

 value of X to the second order is 



c-'n" 



^ = c .sin n{at—x) + —--- .X .cos 2n{at—x). 



If we had treated the equation for waves of water in the same 

 manner, we should have found, to the second order, 



ScV 



X=c.sin n{at—x) a — - . x . cos 2n{at—x)» 



And in both cases, if we carried on the solution to the 3rd, 

 4th, &c. orders, we should introduce the second power of a? 

 multiplying trigonometrical functions of the simple and triple 

 argument, the third power of x multiplying functions of the 

 double and quadruple argument, and so on. The forms of 

 the solutions would be similar, but the coefficients would be 

 different. 



If now for any definite value of x we construct a curve 

 graphically representing the motion of the particles, we find 



that the value of -^ is represented by an unsymmetrical curve, 



the deviation from symmetry increasing as larger and larger 

 values of ^ are taken. The nature of the asymmetry is this, 

 that the interval of time from the extreme backward motion 

 of a particle to the extreme forward motion is less than half 

 the whole period of vibration ; and that this inequality is 

 greater as we consider the movement of particles whose ori- 

 ginal position is more and more advanced. And this happens 

 in the same manner for the particles of air and for the particles 

 of water. 



I might have shown this at once by merely requesting your 

 readers to compare the diagram drawn by Mr. Stokes, in the 



2D2 



