Mr. G. G. Stokes on a difficulty in the Theory of Sound. 351 



curve becomes more and more steep as t increases, while in 

 the neighbourhood of the points o, b, z its inclination becomes 

 more and more gentle. 



The same result may easily be obtained mathematically. 

 In fig. 1, take two points, infinitely close to each other, whose 

 abscissae are z and z + dz; the ordinates will be w and 



. dw , 



W+ -r-dz. 



dx 



After the time t these same ordinates will belong to points 

 whose abscissa? will have become (in fig. 2) z + wt and 



z+dz + (w+ ~ dzjt. 



Hence the horizontal distance between the points, which was 

 dz, will have become 



(«+£')*' 



and therefore the tangent of the inclination, which was-j-, 



will have become 



dw 

 dz 



dw 

 dz 



(A.) 



At those points of the original curve at which the tangent 



diio 

 is horizontal, — =0, and therefore the tangent will con- 

 stantly remain horizontal at the corresponding points of the 



dxso 

 altered curve. For the points for which — is positive, the 



denominator of the expression (A.) increases with t, and there- 

 fore the inclination of the curve continually decreases. But 



when -T- is negative, the denominator of (A.) decreases as t 



increases, so that the curve becomes steeper and steeper. At 

 last, for a sufficiently large value of t, the denominator of (A.) 

 becomes infinite for some value of z. Now the very for- 

 mation of the differential equations of motion with which we 

 start, tacitly supposes that we have to deal with finite and con- 

 tinuous functions ; and therefore in the case under considera- 

 tion we must not, without limitation, push our results beyond 

 the least value of t which renders (A.) infinite. This value is 

 evidently the reciprocal, taken positively, of the greatest ne- 



