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



Take, for instance, the case of a medium in which the par- 

 ticles, being separated by finite intervals, act upon each other 

 solely by their mutual attractions, and in which the vibration 

 is transversal. If in such a medium two waves, analogous to 

 those already described, advance towards each other, the dis- 

 placement in the one lying wholly on the one side of the line 

 of direction of the wave, while that in the other lies wholly 

 on the opposite side, it is evident that at the period of com- 

 plete superposition all particle-displacement must, for the 

 moment at least, have ceased to exist, and all " the tensions " 

 must have ceased to operate. There would, in fact, be an 

 entire annihilation of the disturbance but for the circumstance 

 that each particle, although in the position of equilibrium, will 

 be endowed with a velocity double of that with which it would 

 have been affected if one only of the two waves had reached 

 it. In this case, therefore, the vis viva of the system of two 

 waves, instead of being unaltered, will be doubled. 



If the disturbance in the encountering waves, instead of 

 lying on opposite sides, is in each case on the same side of 

 the line of direction of the waves, it is obvious that the vis 

 viva of the system, which according to Dr. Helmholtz is con- 

 stant, will at the moment of complete occuitation vanish alto- 

 gether. 



II. I now propose to call attention to the singular misnomer 

 involved in the expression "sum of the tensions " adopted by 

 Dr. Helmholtz, which he introduces to our notice in the fol- 

 lowing terms : — 



Taking q for the velocity of a particle whose mass is m, 

 and <£> for " the intensity of the force which acts in the direc- 

 tion of r," it is proved that 



"imd(q 2 )=-cl>dr, 



or, when Q and R represent corresponding tangential velocities 

 and densities, 



^ mQ 2 — ±?nq 2 = — \ <f>dr. 



" Let us regard this equation more closely : we find at the 

 left side the difference of the vires vivm possessed by m at two 

 different distances. To understand the import of the quantity 



1 fair, let us suppose the intensities of </> which belong to 



different points of the connecting line ma erected as ordinates 

 at these points ; then the above quantity would denote the 

 superficial content of the space inclosed between the two 

 ordinates r and R. As this surface may be regarded as the 

 sum of the infinite number of ordinates which lie between r 



