382 Mr. R. Moon's Remarks on HelmhohVs 



positive or negative according to circumstances. If the sign of 



~ in the combination be positive, since it will be multiplied in 



the equation of vis viva by a negative velocity (for the velocity 

 in the combination will be the difference of the velocities of the 

 components, and the velocity in a b c is here predominant), we 



shall have at this point ~ -j- negative ; so that this portion of 



the wave, instead of aiding to balance the negative tensions pre- 

 vailing throughout the front halves of the two waves, as its com- 

 ponents would have done if the two waves had continued sepa- 

 rate, will, so to speak, go over to the side of the latter. 



On the other hand, if at this point of the combined disturbance 



~ be negative, its value will be less in amount, irrespective of 



sign, than what it would have been for the wave a b c taken sepa- 

 rately, at the same time that, as before, it will be multiplied 

 by a negative velocity, but a velocity which will be less than 



the velocity with which — - would be multiplied if we were deal- 

 ing with the wave a b c separately. Hence, though at this point 

 the element of the combined disturbance would, as in the case of 

 each of its components taken separately, tend to counteract the 

 negative tensions of the emerged front halves of the two waves, 

 yet it would do this in a less degree than one only of those com- 

 ponents would do when taken separately (the wave a b c to wit), 

 and therefore, a fortiori, in a less degree than both. 



If we had drawn p P M on the right side of D, we should have 

 arrived at precisely the same conclusion, though in a slightly 

 different manner. It thus appears that, while the negative ten- 

 sions at the time represented in the figure are precisely the same 

 as when the waves were separate, the positive portion of the ten- 

 sions will be diminished in amount, so that they will no longer 

 counterbalance the former. On the whole, therefore, the sum 

 of the tensions, instead of being zero, will give rise to a negative 

 term of finite magnitude in the equation of vis viva. 



III. If it should be supposed that the fact of the sum of the 

 tensions not vanishing in the equation of vis viva may afford a 

 possible source of compensation for the loss of vis viva which it 

 has been shown may arise from the interference of waves, the 

 foregoing investigation will suffice to show the fallacy of this 

 view. For, in the case of motion above considered it is evident 

 that the destruction of the vis viva may be accompanied by the 

 development of a negative term due to the tensions ; in which 

 case there will be a loss of energy, not only through the destruc- 



