Memoir on the Conservation of Force. 383 



tion of vis viva, but by reason of the fact that what vis viva is left 

 will be more or less counteracted by the presence of the negative 

 term due to the tensions. 



It is clear, therefore, that the conservation of vis viva cannot 

 be relied on as holding in cases of the intersection of waves tra- 

 versing continuous masses, and that what Dr. Helmholtz has 

 offered to our attention as a universal law of nature is completely 

 contradicted by fact in cases such as those we have been con- 

 sidering. 



If we advert to the original derivation of the equation of vis 

 viva from the principle of virtual velocities, if we reflect how 

 completely the arbitrary displacement of the points of application 

 of the respective forces forms the characteristic feature of the 

 principle of virtual velocities, and bear in mind that the trans- 

 ition from that principle to the equation of vis viva is effected 

 simply by the substitution of the actual for the arbitrary mo- 

 tions — if we keep in view these various considerations, it can 

 hardly be matter of surprise that, in treating of the internal mo- 

 tions of continuous masses it should have come to be considered 

 that the only terms depending on the internal actions which 

 need be taken into account in forming the equation of vis viva con- 

 sist of pairs of equal and opposite forces multiplied by a common 

 displacement or common velocity (that, namely, of the common 

 point of application of the forces), and consequently that no such 

 terms will appear in that equation. 



That this mode of considering the subject, however specious, 

 must be entirely erroneous, sufficiently appears, if I mistake not, 

 from what has preceded; but it is of the highest importance 

 that the mode as well as the fact of the fallacy should be clearly 

 apprehended ; and to this part of the subject I propose now to 

 address myself. 



Let p v p 2 be the pressures, and v v v 2 the particle-velocities 

 at the time t at the surfaces the ordinates of whose positions of 

 rest are respectively x + dx and x + 2dx; p_ Xy p_ 2 , V-i, v_ 2 the 

 corresponding quantities for the surfaces as to which x—dx 

 and x—2dx are the ordinates of the positions of rest. 



The equation of motion of the first element may be put under 

 the form 



0=T)~dx+p 1 -p; 



multiplying which by vdt we get 



dv 

 0=Dv -j-dtdx+PiVdt—pvdt. 



The corresponding equations for the elements in contact with 



2D2 



