502 Mr. Tovey’s Lesearches in the 
& = d.asin (nt—kxv+a)+.Bsin (nt+ha+0), 
y = X.a, sin (x,t—k, x+a,)+=.B6 sin(nt+k r+6), (2.) 
¢ = 2.a,sin (nv, f—hk,v+a,)+=.6,sin (n,t+h,a+5,); 
where 
WS PAS I eH eT Se.) 
n = WV (spk?P — Pk} + &c.), (3.) 
my = V (shy sky + &e.): 
the sums = being extended to all the requisite values of the 
arbitrary constants. 
We perceive, by the equations (2.), that the motion of the 
system may be regarded as compounded of a number of co- 
existing movements, severally expressed by the terms of the 
sums =. And when we confine our attention to a single term 
of the first sum in one of these equations, which we may do 
in a great variety of problems, we have virtually the same ex- 
pression for the displacement of a molecule of an undulating 
medium, as is assumed tacitly by Sir Isaac Newton, and ex- 
pressly by Professor Airy*. 
Taking separately the displacement y, and considering only 
one term of the first sum in the expression for this quantity, 
we have 
; 7 = a, sin (n,t —k,x + a). 
It is well known that this equation represents a series of 
equal and continuous waves, the length of each wave being 
25 g : 5 : 
pre where 27 is the circumference of a circle whose radius 
ic 
éf ~ . 
is unity. Now, if we increase x uniformly, so as to make 
k hnctaie : a dz. 1, 
mt—k,« constant, remains constant, and > = ee 
Hence we perceive that these waves travel, in the direction of 
2 positive, with a velocity equal to aps If »= 8, sin(n é 
+k,x+ b,), which is a term of the second sum, the move- 
ment is similar, except that the waves travel in the contrary 
direction. 
The second of the equations (3.) gives 
is s/? 5 
%; = 5,4 (1- aera + &c.) 3 
an equation affording the same theory of dispersion as that 
which has been so satisfactorily investigated and verified by 
* See Airy’s Mathemat. Tracts, p. 259. 
