Theory of the Physical Forces. 449 



According to principles and processes therein indicated, it 

 appeared that the determination, purely by analytical reason- 

 ing, of the rate of propagation in a homogeneous medium the 

 pressure of which varies proportionally to its density requires 

 the integration, at least approximately, of the equation 



« + 'i^,--0. ..... (B) 



To effect this integration, the form of/ was assumed to be such 

 that we might have 



fz=cr~i cos (mr + </), 



c, m, and d being certain constants. By substituting this 

 value of/ in the equation, a result is obtained differing from 

 zero by the residual quantity 



(i-» 2 + 4*)/- 



Hence it follows that, whatever be m, the equation is exactly 

 satisfied by all the special values of r for which /= 0, which 

 are the special values that cause cos (mr + c f ) to vanish. Now 

 it is evident that the value of the residual quantity is coeteris 

 paribus least when m 2 = 4te, and hence that the approximation 

 to the solution of the equation (B) is closest when m = 2\/e. 

 This, in fact, is the case in the approximate solution which I 

 referred to as having been obtained by Sir W. Hamilton and 

 Professor Stokes, whence I deduced a velocity of sound ex- 

 ceeding by 17*5 feet the experimental determination. In the 

 above-mentioned postscript I made the supposition that m — ir, 

 and consequently found a velocity of sound only 3'17 feet less 

 than the observed value. But although this difference is less 

 than the other, that supposition must be rejected, because the 

 equality m 2 = ir 2 e does not give so close an approximation to 

 the solution of the equation (B) as ra 2 = 4e. Hence I have 

 finally come to the conclusion that the principles and processes 

 of Hydrodynamics which I have adopted conduct, by pure 

 reasoning, to a velocity of sound exceeding by 17*5 feet the 

 latest determination by experiment and observation. Possibly 

 the excess may be attributable to the fact that the air, instead 

 of being homogeneous, as the theory supposes, is composite, 

 and in its lower strata charged with vapour. 



(2) The assumption at the top of page 29 of the January 

 Number, that (alty) has the form (cZ ./</>), /being a function 

 of x and y only and (p a function of z and t only, may appear 

 to be arbitrary, because I omitted to infer from the equation 

 (A), p. 27, that (cZi|r) = if X=l, and the propagation be 



