ANALYSTS TO THE DYNAMICAL THEORY OF THE TIDES. 
221 
Hence, provided li be constant, the two members will be identical, if for all positive 
integral values of n we have 
C«_2 
{2n - 3) {2n - 1) \ tz {n + 1) ^ (2??. - 1) {2n - 3)/ {2n + 3) {2n + 5) ~ 
C 
( 22 ), 
it being understood that Og = 0, = 0. 
This is on the hypothesis that the depth is constant ; a more general hypothesis 
would be to suppose that h is of the form 
-f- ^(1 — 
where h and I are constants 
Assumnng this form for h, the left-hand member of (21) becomes 
ni - Sr. (1 - sr., (1 - . 
which, by the properties proved above, is equal to 
{n-2){n-l) 
_(2yi - 3) {2n - 1) 
r._,, - 
2n{n + 1) 
{2n - l)(27y + 3) 
^r. + - 
(w A 2) (n 3) 
{2n -f 3) (271 + 6) 
dVn 
Identifying this with the right-hand member, we obtain 
C„_=, . / P -1 . 2 \ . C 
(2??, — 3) (271 — 1) 
y; + 7R 
71 (71 + 1) (2/1 — 1) (271 + 3)/ (2?! + 3) (2n -|- 5) 
n+<2 
ITn I r (71- 2) (71-1) 27l(71-fl) (71-I-2) (71+ 3) 
2 /O,. 1 \ , O,, 1 ON ~t“ ^ 
4&)^rd ‘Iw-P l(2?i —3)(27 i—1) 
(271-1)(27i + 3) 
(271-1-3)(27i-1-5) 
n +2 
(22 a). 
On introducing the values of r„ from (19), equations (22), (22 a) may be written 
c„_2 (' r - 1 2 /i^. 
(2?i - 3) (27? - 1) 
(71 -1- 1) (271 — 1) (271 + 3) 4a) 
_ ^>7,1 
{2n -f 3) (271 - 1 - 5) 4ara? 
. . . . (23). 
C,,_, 
1 - (71 - 1) (71 - 2) If/n-il4a)ht~ 
(27? - 3) (27? - 1) 
(/^ - 1) , 2(1- 71 (71 + 1) Iffnpaf^ 
71 (71 + 1) (271 - 1) (271 4- 3) 
hn 1 
4&)%^ J 
c. 
^+2 
J 1 — (71 -f 2) (71 -f 3) fffa+zl4o)hd 1 
I (271 + 3) (271 -f 5) j 
H, _ I r (71-2) (71-1) 
4a)%2 4co~P L ( 271 - 3) (271-1) 
7/7-2 
2 ?! (71 4 - 1 ) 
(271—1) (271-1-3) 
(714-2) (714-3) 1 V 
3) 7" + (2n+-6)(inT5) J ’ 
