186 Proceedings of the Koyal Society of Edinburgh. [Sess. 
where k is positive, by Cauchy’s Theorem of Residues, taking as contour 
the real axis of /x together with an infinite semicircle below the real axis, 
we obtain for it the value 
in 
r=-co -a - rw) 
w 
Similarly if we evaluate the integral 
X IX m sin — (a - tx- ik) 
IV 
taking as contour the real axis of /x together with an infinite semicircle 
above the real axis, we obtain for it the value zero, since the integrand has 
no poles inside this contour. 
Subtracting the latter result from the former, we have 
t//x/(/x) 
sm—{x-fx-ik) cos— {a- IX- ik) _ ^ ^^(x-a-rw) 
x-ix — ik • 7T / X / \ 
^ sin — {a - IX- ik) “ — (x -a- riv) 
w iV 
Similarly we have 
2TTi 
sin — (x-fx + ik) cos — (a-ix + ik) p , 
, ,, X 10^ ^ ^ ^ ’ 1 ^f{a^-riv-^ik).e 
^-{x-a~rw) 
(lixflix)- 
X fx + ik] sin — (a - /X + ^’A:) 
— (x - a - riv) 
w 
and thus we obtain 
G(x,k)^^ 
iTT In 
f(a + 7nv - ik) . - f(a + rw + ik) . e 
: 2i—(x — a- riv) 
w 
so that 
|lim jfc_^o k) = 
00 f{a + 7'w) sin — {x - a — rw) 
•?oo ^ 
— (x -a- rw) 
w 
Now (a{x, k) is the function which was formed from g(x,k) by replacing 
all the short-period terms by the corresponding cotabular long-period 
terms: and (as in Poisson’s discussion of Fourier’s integral) we have 
/(o;) = lim&^oP'(a;, k ) . 
Hence we infer that the expression 
00 
E 
r= — 01 
f{a + rw) sin —{x -a - rw) 
—ix -a- riv) 
w 
(3) 
