NATURAL PHILOSOPHY. 183 



lyrically allowable, is not allowable in the problem of sound is, that in that 

 assumption quantities are neglected which (in the case of sound) are of the 



d'-ii ff-i/ 



same order as those which are retained, so that the equation ^- == ,u 



fJt- (JX& 



is not an approximation, but is reduced to the form = 0; from Avhich noth- 

 ing can be inferred but that the assumption f - -j = 1 is not admissible. The 



result of this reasoning is, that the equation y = F (x -\- at) -\-f (x at), which 

 has hitherto been the basis of explanation in Treatises on Sound, has noth- 

 ing whatever to do with sound, but represents the motion of a wave in an 

 imaginary elastic medium of a constitution the very opposite of that of the 

 atmosphere and of all known gaseous media. The mathematical theory of 

 sound is consequently put back to its differential equations, beyond which 

 not an inch of ground can be maintained. Till the differential equation is 



integrated accurately, without assuming (-j-)= 1, no advance can be made, 



and science remains u-ithout a mathematical theory of sound. The author 

 then announced that he has succeeded in integrating the differential equation 

 of sound without approximative assumptions; that he has, in fact, obtained 

 its exact integral ; and in the result has possessed himself of the key to the 

 various properties of sound. Among several others, it was stated that the 

 exact integral accounts for the great difficulty which experimenters have 

 found in obtaining accordant velocities of sounds, for the sweetness of mu- 

 sical sounds, for the rapid decay of violent sounds as they progress, and 

 proves that the velocity with which a sound is transmitted through the at- 

 mosphere depends on the degree of violence with which it was produced, 

 and not (as in light) on the length of the wave ; so that sounds of every pitch 

 will travel at the same rate, if their genesis do not differ much in violence; 

 but a violent sound, as the report of fire-arms, will travel sensibly faster than 

 a gentle sound, such as the human voice. This last property the author 

 stated to have caused him much trouble, in consequence of its being directly 

 opposed to the testimony of almost every experimenter. For many affirmed, 

 as the direct result of their observations, and others assumed, that all sounds 

 travel at the same rate. Fortunately, it transpired at the meeting that in Capt. 

 Sir J. Franklin's Expedition to the North, whilst making experiments on 

 sound, during which it was necessary to fire a cannon at the word of com- 

 mand given by an officer, it was found that the persons, stationed at the dis- 

 tance of some miles to mark the arrival of the report of the gun, always 

 heard the report of the gun before they heard the command to fire; thus 

 proving that the sound of the gun's report had outstripped the sound of the 

 officer's voice; and confirming in a remarkable manner the result of the 

 author's mathematical investigations. 



The Astronomer Royal said, that while he had no doubt whatever of the 

 general accuracy of the conclusions at which Mr. Earnshaw had arrived, and 

 while he fully admitted their importance, he could not subscribe to all that 

 he had said. In his historical sketch of the steps by which we had arrived 

 at our present knowledge of the subject, he could not admit that the method 

 of Newton was wrong, the fact being, that it was a strictly correct solution 

 of one case of what Avas a very general problem. The method of Newton 

 was the very basis of all our modern methods ; and he looked upon that por- 

 tion of the second book of the Principia as a monument of the genius of 

 , which he was very sorry to see was beginning to be much less 



