A SYSTEM OF PARTICLES. 241 



from which it appears that if w = 2 our expression is — a- ^-^ — 5 an 



essentially negative quantity. This conclusion was seen to be requisite, 

 in order to satisfy the conditions of direct vibrations with repulsive 

 forces. 



It is also evident that the same conditions would be fulfilled by 

 making n greater then 2, whereas if u be zero or negative tliis will 

 not be the case. The hypothesis, which makes n equal to unity, we 

 shall examine hereafter. 



5. By integration of the equation (1) and reduction, we obtain for 

 the square of the velocity of transmission, 



" = 2'^-' r- ' ■\—r~/^ — n^' — \~^/' 



which, if 6 be put for the distance between two consecutive particles 

 of ^, and E for that between two of B, and we make 



Sx = ^e, Sy = )je, Ss, = ^f. 



Ax = XE, i\ij = YE, As = ZE 

 becomes 



*^P^M •"''-'''' ... ^^■- ^^■^■- ))• 



^^ ^ \X^+Y^+Z')~ '^^ ' 



For the present, we will omit that part of the expression which 

 depends on the length of a wave ; hence we have 



\v + 'f + V)~ \x-' + Y + z-')— ' 



