ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 659 
From (18) we conclude that 
PENG ~ 4) 
Cos, (1): = — 
a Tm Tn (20) 
Shea ene ges) 
Sin, (1) = 1 oT@)TG- 4) 
a 5 
*r(i-%-a)r(1+ 2) 
Hence 
T (T4))? 1 A 1 a T 
, — = = H T a — 2s 
Cos, (1) COS % 5 i ae G ane i 5 4 
5 +5) COS = 7 
bo] 
| 
we vols 
Sa Se 
| 
Sin, (1) = *sinn® 
ll 
3 
=] 
Game 
pen 
| — 
iz 
a aA aS 
eS bo 
aN 
ts| = 
Ney 
| 
_ 
i 
bo| = 
Nee 
aw 
b| 3 
NS 
| 
ES) 
dS! 3 he 
which are well-known relations between the [-functions. 
From (20) we see at once that 
Cos,(1) = 0, if n(n+2a) = 1.(1 — 2a), 3(8- 2a), 5(5-2a)...... 
Sin,(1) = 0, if n(w+ 2a) = 2(2-2a), 4(4-2a), 6(6 -2ay)...... 
Applying this result to the Seiche-functions, we find 
Cask, (e l= 0 fore 17 dA, t,o 
Sime (Gel) =" 0 forie’ = "270i, 4.0 6.7, 2. 2. 
[f in equation (3) 1 represents a positive integer, the functions C,“(w) have a 
peculiar significance. They are then the coefficients of the powers h” in the series 
(1 — 2h +h®)-* = S°A"C,"(w) . 
0 
Now, according to an important theorem 
dt 
Waae -" Qari 4 (t= wy ) 
ChAKOD) Te)! i F(t) 
if f(t) represents a function which is regular within the contour C (see WHITTAKER, 
Modern Analysis, p. 53). Since we had before (by (13) and (14)) that 
(1-#)"+¢-4d¢ 
c (¢—wyts 
(1 — Bere’ 
e G=ny ~~ 
C,,"(w) = const x | 
= const (1 — w*)-* 
we notice that C,”(w) is proportional to 
Qrt2a-1 
aes { (1 ae ape yeta=t ; 
and also to 
(le aa | (1 — wy \ ; 
We verify without difficulty the following relations :— 
a(p) =( —9)r Het (at 2)... . (@+n-1) op fe Solis ee, 
ee Gemeente) ae 
= — 9)" a(a + 1)(a+ 2) Ce TO (a+n-—1) ae ie }= ory, D) 
[) Mem cai eee ae” em 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART III. (NO. 26). 97 
