306 Prof. Powell’s further Observations on M. Cauchy’s_ 
of motion deduced upon M. Cauchy’s principle (in my ana- 
lysis, eq. (12.),) will be reduced to 
f (7 os? ? 
Fh HS { mith) 4, Ve (1.) 
where 7 is the value, at the time ¢, of the varying displacement 
of the molecule m, whose rectangular coordinates when in 
equilibrium are x y z34+Avy is the displacement at the same 
moment ¢, of another molecule m, which has for its rectangu- 
lar coordinates when in equilibrium 
rt+Ac ytAy z+Az, while 
r= V/ Aa’ + Ay? + Az’, 
or the distance between these two molecules in their positions 
of equilibrium; £8 is the angle between this distance r and 
the axis of y; and, finally, f(r) and f(7) are functions of 7, of 
which the former (if positive) expresses the law of attraction, 
or (if negative) the law of repulsion, and the latter is derived 
from it by the rule 
FitD = 7 P zr). —T(7- 
S, the sign ef summation, is relative to the actions (attractive 
or repulsive) of all the molecules m. 
I have recapitulated thus far in reference to what was esta- 
blished at the outset of M. Cauchy’s investigations. “Now this 
analysis is thus far devoid of all difficulty or intricacy ; the 
whole difficulty of the subject lies in the zntegration of these 
equations of motion. The integration given by M. Cauchy 
is of an extremely general kind: but for the purpose we have 
now more immediately in view, it will be readily allowed that 
if a particular solution were proposed, such as to include the 
establishment of the relation between » and A, it would suffice. 
A valuable instance of a method of effecting such a simplifica- 
tion has been laid before the readers of this Journal, in the 
excellent paper of Mr. Tovey in the Number for January, p. 7. 
But another such particular solution has been pointed out 
by Sir W. R. Hamilton, the nature of which I now proceed 
to describe ; and this will be most perspicuously done in the 
following manner: 
It will be easily seen that all the conditions of a wave for 
the ordinary phenomena are fulfilled by such a function as 
4 = A+Bcos (<7 we-) +C sin (= (we—t)); (2) 
