632 PROFESSOR CHRYSTAL 
Kliminating A, and B,, we get finally 
As {@AC(c, , 1) + avS(e,, 1)}{ag'w'C(e,' , 1) + ay‘p’S(e,’, 1)} 
+ Ag{dyN'C(ey’, 1) +.a,'v'S(cy, 1)} {aguC(e,1) + a,pS(e,,1)}=0; . : (49) ; 
which is the period equation for the lake. When the lake is symmetrical, that is, 
when @,=(;', @,=,, ete., this equation simplifies, and breaks up into the two 
following :— . 
AyO(C, , 1){a,8'(Cy, W)C (Cg , 03) + AyS(Cy 5 Wy) E'(eg » Wy) } j \ | 
+ a,S(e,, 1) {ag0'(ey , we) O (eg, Wg) + A.C (Cy 5 Hy) B'(cy, w,)} =0, | ; 
and AO(C,, 1){4g5'(ce , Wy) S(Cy , Ws) + AyS(Cy , Wy) S'(Cy » Wa)} i (ay 
+ a,S(¢,, 1){agC’(e,, W_)G(cg, Wg) + AgC(Cy , My) S'(cy , wz)} =O | 
ALTERNATIVE SOLUTION FOR PARABOLIC LAKES. 
INTRODUCTION OF THE LAKE FUNCTION. 
§ 41. For certain purposes a modification of the solution for parabolic lakes is 
convenient. This is obtained by shifting the origin to the positive end, and, for 
convenience, halving the scale of the new variable; that is, we put w=1—2z. The 
equation 
(1- ALE P= 
dw? ; 
then becomes 
Ga PaO . oon 
= ; 
If we attempt to solve this by the assumption 
PHA + AyztAge?+ -- +--+ 5 
we find in the usual way that we must have 
A=0, 
cA, +1.2A,=0, 
(c-1.2)A,+2.3A,=0, 
(c—2.3)A,+3.4A,=0, 
(c_w=2.m—1)A, j+n—1.n7A,=0. 
‘Therefore A, = —A ees 
=(-1)" ic(e— 1.2)(c-—2.3) . ‘ (e=m=2.n=1)y | 
eRe eee ish 
‘That is to say, we find 
&: Ce we—laye cele mee eee ) 
oP oe Soe a | 
where the series within the brackets is obviously convergent for all real values of 2 
between —1 and +1, both included. 
