34 On Newton's " Regula Tertia Philosophandi" 



propagation of sound, which, as stated in the foregoing com- 

 munication, was obtained in the Philosophical Magazine for 

 May 1865 (p. 329), and introduced into the ' Principles, &c./ 

 prop, xiv., pp. 214-224, was derived from an exact integral of 

 the equation 



namely, 



/= (4rim/e)~* cos ( 2rs/e- \ J • 



f 

 I argued that the term —- 2 might be omitted for very large 



values of r, as being incomparably less than the other terms 

 by reason of the denominator Ar 2 . It has, however, very re- 

 centlv occurred to me that under the same circumstances the 



term -~- is incomparably less than either -f{ or Aef, as might 



readily be inferred from the given expression for/. The in- 

 tegral I employed was therefore an approximation, for large 



d? f 

 values of r, to that of -f^ + 4</= 0, but not, as I supposed, to 



that of dr 



d2 f ■ : d f , a * n 

 dr 1 rdr J 



In fact it may readily be shown that the latter equation is 

 exactly satisfied by the solution, 



f=^ CO s[7rr^e-fj, 



for the special values of r which make the cosine vanish, and 

 for all other values with increasing approximation as r is 

 greater, and that these special values are separated by the 



constant interval — t=. These results confirm those obtained 

 \/e 



by the very different investigation I gave in the Number of 

 the Philosophical Magazine for February 1853, which I was 

 induced to relinquish only under the above-stated misap- 

 prehension as to the applicability of the former expression 

 for/. After full consideration I have not been able to discover 

 any fault in the reasoning of that investigation. The rate of 



/ 4 V 

 propagation it gives is a 1 1 + — 2 1 , which, if we take the value 



of a to be 916,322 feet, as adopted by Sir John Herschel, we 

 obtain for the velocity of sound 1086"25 feet, which is only 



