134 THE EEV. S. EAENSHAW OX THE IMATHE^IATICAL THEOET OF SOUXD. 
On comparing (3.) with (2.), we find F' 5 oiS for breTity, writing a for 
we have 
and 
OL 
••• F(a) = C±>s/^.log,(«). 
But as ^=F(a) from (1.), it follows that 
dii=%-dx+%.it 
= a(Z^+F(a).(?^, 
which being integrated in the usual manner, substituting at the same time for F(a) its 
value, gives 
y—(xx +(C +\/ ^ log,ci)t+(p{a) ] ^ ^ 
0=cix +\/ (jjt-\-oi(p'(a) J 
Between these equations, if we eliminate a, we have then the integral of equation (3.). 
2. From equation (4.) we see that a=^ ; and if we represent by u the velocity of the 
particle whose place is we find 
«^=^=C+x/j(Alogea, 
= C+>/ ^log,^^^ ; 
u — C 
and 
3. To determine the arbitrary constant C, we observe that f=fo u = C ai’e alwavs 
simultaneous equations. But the former belongs to the confines of the wave, where in 
fact w=0 ; and therefore C=0. Hence for a wave transmitted through a medium which 
is itself at rest beyond the limits of the wave, we have these equations*; — 
^ = (6.) 
0=a.r +\/ yjt-\-a(p'{oi) j 
* If a; and a be eliminated between the equations (7.) and u= + logi a, we shall obtain the equation 
“=/{5'-(“+ 
which was first obtained, though in a very different manner from that employed in this paper, by M. Poissox, 
and printed in the Journal of the Polytechnique School, tome vii. It seems not to have occru’red to hira, 
however, that by means of this equation he might effect another integration of the equations of fluid motion, 
and thus discover the relation between p and u, whereby his solution would have been completed. 
Several of the properties of wave-motion, depending on the gradual change of type, which are included 
in this equation of M. Poissok’s, were first brought forward and discussed by Professor Stokes in the 
Philosophical Magazine for November 1848, and by the Astronomer Eoyal in June 1849. In the latter 
