i 
614 PROFESSOR CHRYSTAL i 
The amplitudes being small, we shall neglect quantities of the order of ¢,(dé/éz)? 
The above equation then becomes 
b(x)t= ae -#)—A(@)-é 
ait —eh@), 
eA) 
ax” 
or 
f-—apli@} .  .  . nn 
and 
A(x p= ag Aa) : : . 
If we substitute the value of ¢ in (4), we get 
A(z) oe = 9Ale) 2 Eee {Awe} | 
Now this last equation may be written 
oe SEES He) b( = dx 5 | ie) (x) att e yet] ; oF 
If we determine new variables u and v by the equations 
u= Kee) b= [xt e) Co 
then (5) may be written 
Cea gAla 2) b(a 21 2? = go Om 9 . . = (7), 
where «x is to be determined as a function of v by the second equation of (6); and o(v) 
=A(xr)b(x). Also (3) becomes 
f=-— . : i . 
The curve whose ordinate and abscissa are « and v we shall call the normal curve of 
the lake. 
Since a seiche is a standing oscillation, £, and therefore wu, is a periodic fanceee of 
the time. We may suppose this periodic function analysed into simple harmonic terms, 
and write 
u= SP sin n(t - 7) , é : ‘ : (9), 
where P is a function of v alone and 7 is constant. The values of n admissible depend 
on the circumstances of each case: but, in order that (9) may satisfy (7) we must have 
2 
-v?P= go(v)- = : 
The mathematical theory of a seiche of small amplitude depends therefore essentially 
on the differential equation 
; : (hell eon 
= . . . 1 » 
dv tg Ci ; ‘ (10) 
where «x is determined in terms of v by the equation v= | dxb(x); and o(v) = A(a)b(a). 
