ON THE HYDRODYNAMICAL THEORY OF SEICHES. 633 
We have thus obtained only one synectic integral of (50). A second integral may 
be derived in the usual way, but it is not synectic. It has, in fact, an essential 
singularity at z=0; and may be expressed in the form P logz+(z), where \(z) is a 
power series. It is needless to state the actual result here, as it is of no use for our 
present purpose. 
Dropping the multiplicative constant, we write 
2 5 he 2) 23 plintes =) gt 
C7  de= il )z _ ele: 1.2)(c <3) —+ Bee ey : ‘ (51) ; 
L Se a 
(.2)=*— 9+ T9e2 3 129? 32 
or 
73 
2 : 
EC ceo — e(1 -¢/1.2) 3 
at 
— c(1-e¢/1.2)(1 - ¢/2.3)s i 
mae a 6) en eiB: (6,6 - en = Fs a (51’). 
The function thus defined will be called the Lake Function. It is obvious that 
, P—L(e,2) 
is that synectic integral of the equation (50) which vanishes when z=0, 1.e. when 
w=1; and it will be valid for 15z 0, that is, for -1<w<+1, in short, throughout 
the whole length of the complete parabolic lake. 
In particular, we have 
or 
Om (a-S)a-s3)(0-24) aera . | yee 
by EuiEr’s Identity. 
Hence, again, we get 
Le(Crl rae (ead) Scar, I). | 
(53), 
2 7 Ge 
- = cos | 5 J(4e+ ie 
if we use the I’-expressions for C(c, 1) and S(c, 1) given in § 25. 
Since L(c,z) is a synectic integral function of the general equation for P, which 
has the two synectic integral functions C(c,w) and S(c,w), that is, C(e,1—2z) and 
S(c, 1 —2z), it must be possible to determine two constants A and B, so that 
L(c, z) =AC(c, 1 — 22) + BS(c, 1 — 22). 
Thus, in particular, we must have, forz=0 and z=1, 
AC(c, 1)+BS(c,1)=0; 
NC(c, 1) BS y= ie, 1)= Cle, Se, 1) 
Whence it appears that A=48(c,1), and B= —4C(c,1). Therefore 
2L(c, 2) = S(c, 1)C(c, 1 — 22) — C(e, 1)S(e, 1 — 22) ; 
or oe 
| Sy 
aL(c, ) =8(c, 1)C(c, w) - Oe, 1)S(c, w). 
2 
