116 Mr. R. Moon on Helmholtz's Memoir on 



sum of the tensions " (= j otdt taken between such values of 



t as may be fixed upon) will not vanish, as is tacitly assumed 

 by Dr. Hclmholtz. 



Moreover, multiplying (1) by ■—, and integrating with 



respect to a, we shall have 



D [dx&- + faf-f = Action of t ; 

 J dt 2 dt J dxdt 



and therefore, integrating with respect to t, 



^ LM+ {dt {dxff = function of *. 

 % J dt\ J J dxdt 



Hence, if the integration be made between the epochs t 1} t 2 

 occurring during the period of interference, the right-hand 

 side of the equation will not in general vanish, from which it 

 results that in the case before us, during the period of inter- 

 ference, the sum of the vires vivce and of the tensions will not 

 vanish, as asserted by Dr. Helmholtz *. 



We have hitherto considered the results which occur when 

 both the encountering waves are condensed. When both the 

 waves are rarefied, the effect will be precisely similar; but 

 when one of the waves is condensed and the other rarefied, a 

 material variation in the phenomena occurs. 



We will suppose, as before, that the waves are of the same 

 length, and have each a single maximum, but that at the 

 middle point, and at equal distances from the middle point, 

 the excess above the mean density in the one is equal in 

 amount to the defect below the mean density in the other. 

 In this case the velocity of each particle of the one wave will 

 necessarily be in the same direction as that of each particle of 

 the other. 



Under these circumstances, at the moment of complete 

 occultation the condensation at each point will disappear, but 

 the velocity at each point will be doubled ; so that the vis 



* In my former paper I showed that when, the period of complete 

 occultation being passed, the overlapping is confined to the hinder halves 

 of the waves, the function a will be negative, whence I too hastily drew 

 the inference that the last term of (1) will be negative. In point of fact 

 in the same way in which it is shown that when the hinder halves only 

 overlap the function a will be negative, it may equally be shown that 

 when the front halves only overlap the same function will be positive. 

 In fact a, which will be zero at the beginning, middle, and end of the 

 interference, will be positive during the first half, and negative during 

 the second half; and the sign of the last term of (1), the integration 

 with respect to t being taken between epochs which occur during the 

 interference, will vary according to the particular epochs we may select. 



