Memoir on the Conservation of Force. 381 



out the hinder (left hand) half of the wave, and negative through- 

 out the front half; and as the velocity in A B C is throughout 

 positive, it follows that throughout the hinder half of A B C we 



shall have -J- -j- positive, while throughout the front half of ABC 



tltO Lit 



the same function will be negative. For the entire wave, there- 

 fore, it is clear, from the symmetrical form of vibration which 

 we have ascribed to it, that the sum of the tension will be zero*. 

 If we now consider the wave a b c taken singly, we shall have 



throughout its hinder {right hand) half -j- negative, while 

 throughout its front half ~ will be positive; and the velocity 



throughout abc being negative, we shall have -J- -M positive 



throughout the hinder half of the wave, and negative through- 

 out the front half. 



Thus, in the case of either wave taken separately, the positive 

 and negative parts of the term in the equation of vis viva depend- 

 ing on the tensions will counterbalance each other, and the ten- 

 sions will wholly disappear from that equation. But when the 

 waves are superposed, or interfere, as, for instance, in the man- 

 ner represented in the figure, this mutual balance of the oppo- 

 sing terms will cease to exist, as I shall now proceed to show. 



The figure is supposed to represent the state of things occur- 

 ring after the period of complete occultation, when the front 

 half of each wave has entirely emerged, and while the hinder 

 halves have in part emerged, but as to the remainder are still in 

 the state of superposition. 



Draw p P M perpendicular to c A C, and on the left of D, the 

 point of intersection of the curves ABC, abc. 



When the waves are separate and non-interferent, a stratum of 

 the hinder half of either wave of the undisturbed breadth dec, in 

 the case of the one corresponding to PM, in the other to p m, 

 would give rise to a positive term in the equation of vis viva. 



But when these elementary portions of the two waves become 

 superposed, although the condensation of the combination will 

 be the sum of the condensations of the two elements taken sepa- 

 rately, yet, inasmuch as the values of — for the separate elements 



have opposite signs, the sign of -~ for the combination may be 



* Though less obvious, the same is equally true whatever be the form 

 of vibration, provided that the wave is such as to be transmitted without 

 undergoing change in its length, or form of vibration. 



Phil. Mag. S. 4. Vol. 49. No. 326. May 1875. 2 D 



