Undulatory Theory of Light,—continued. 501 
& =psinke + p'sinkax + p’ sin kle + &e. 
+ qgcoskx + q' coskx + gq" cos k"x + &e. 
where Ps P's p", &es q 75 7", &e. are functions of t, and hk, k!, k", 
&c. arbitrary constants. (Poisson, Traité de Mécanique, 
No. 514.) Now, if we substitute p sin kx + q cos k x for = in 
the first of the equations (1.), it becomes 
a 
{art (@#-SH +8) p | sinks 
et eR seis ie ka = 0; 
rr: ( -) 7p coska = 0; 
and, this equation being true for all values of x, resolves itself 
into 
dp 212 2 
oT tae (s ke —s” fA &c. ) p=0, 
dT . (ej2 gh 
apt (s i? —s! i++ &e. ) iO. 
The complete integrals of these equations are p = A sin né 
+Beosni, g = A'sinnét + B' cosné¢; where A, B, A’, BY, 
are arbitrary constants, and x = 4/ (s? 4°—s!? k*.4+ &e.). Hence 
the complete integral of the first of the equations (1.) can be 
expressed by the sum of a series of functions, each of which 
is of the form 
(A sin x¢+B cos n#) sinkx+ (A’sinnt+ BI cos nt) coskx. 
This expression is, by the rules of trigonometry, equiva- 
lent to 
A+B! A'—B 
2 2 
cos (zt—kax) + sin (nt—k x) 
A-—B ) 
d A 
a ae Bie (mt+hk ax) + E svenepe 2) 
which, if we put 
A+B! s A'—B : ’ 
: = asin a, —Z— = @ Cos a, = — Bsind, 
'4+B 
ee = pcos 0, 
may, by the same rules, be reduced to 
asin (nt—ka +a) + Bsin(nt + ke + 0). 
It follows, therefore, that since the second and third of the 
equations (1.) are of the same form as the first, and since » 
and % are, like 1, functions of 2 and ¢, we can express the 
complete integrals of these equations by putting 
id 
