ON THE PERIODS AND NODES OF LOCHS EARN AND TREIG. 831 
To facilitate calculation we introduce the following notation :— 
Cie) a Mons — Rona 
where Le se ee aoa Once ell) (13) 
IN Rae 2l Glee CR) ele ater (1 —¢/(2n—1)2n), ’ 
A aoe 
SC — ees =f, — Ran: 
where Don = Aan | 200, (14) 
eel C) 203). a se (1 —¢/(2n — 2)(2n—1)), : 
A, =c 
If we take n so great that c/(2n—1)2n or c/(2n—2)(2n—1)<1, then 
(Re asen I< | Arnss | wr 3(] + w+ wit ad oo); 
2n+1 n+3 o« « » © > 
2n+3 1—w" 
That is, 
| Ron +1 |< | Tones | /Q = w?) . . U : O o (15). 
In like manner, 
Ee eaneee (iia seen tte Ey TU wl P trode Sate), 
Similarly, if we write 
Wigan = 4 Re 
where ihe = B,2"/n ’ (17) 
Beco)... 2. (1-1/(n—1)n), : edi 3 
B,=¢, 
then we can show that 
etea aul (ea cy Ae eo MUGS): 
The formulee (15), (16), and (18) enable us to estimate the accuracy of the approximation 
obtained by taking any given number of terms of the respective series. It is obvious 
that the formule (9), (10) are most convenient for nodes near the deepest part of the 
lake ; and (11), (12) most convenient for nodes near the ends. In most of the calcula- 
tions tabulated below the equation L/(c, z)=0 is used; but in many cases we verified 
our results by working with the other formule as well. Other things being equal, the 
formule (11), (12) have an advantage, in respect that tabulation of the steps for the 
calculation of B is one continuous operation, and there is less chance of error by in- 
advertence in the entries. 
§ 12. As for the periods, so also for the nodes we get first approximations by taking 
the mean between the extreme cases of a complete symmetric parabolic lake and a 
semiparabolic lake. As might be expected, these first approximations are not so close 
for the nodes as they are for the periods. Fortunately the series are very manageable. 
If we denote the vertical seiche displacements at the Western end, deepest 
TRANS. ROY. SOC, EDIN., VOL, XLI. PART III. (NO. 32). 122 
