OF MOLECULAR FORCES. 105 



parallel to the same axis. We may therefore consider the effect of each 

 component of the displacement separately. 



It appears from the equations just obtained, that — d)V, -— (P g V, 

 — d\V are the absolute forces of restitution. 



Since one at least of the quantities d) V, d] V, d\ V is negative, and 

 one at least positive, there will be at least one principal axis parallel 

 to which a disturbed particle can vibrate, and at least one parallel to 

 which a disturbed particle cannot vibrate. Suppose for instance, that 

 d}V is positive and d\V negative, then the first equation dfx = d}V. x 

 takes the form 



dfx — a 2 X, 



the integral of which is 



x = CV' + C"e- at ; 



a result which shews that x must increase continually with t. The 

 motion in this direction will therefore be one of translation. 



But for that part of the displacement which is parallel to the axis 

 of %, the equation of motion is 



d?% m — 7 2 s. 



The integral of which is 



% = A cos (7/ + B), 

 which denotes vibration. 



13. If the constitution, or arrangement of the particles, of the 

 medium is such that d g V is positive, the motion parallel to y will be 

 one of translation; and consequently there will only be one line in 

 which a particle can be displaced, so that its motion may be vibratory. 



14. If the constitution of the medium be such that d* g V is negative, 

 the motion parallel to y will be vibratory; and therefore if the particle 

 be displaced in any direction in the plane y%, it will continue to vibrate 

 in that plane, describing an elliptic orbit. 



Vol. VII. Part I. O 



