638 PROFESSOR CHRYSTAL 
If ,v, denote the distance from O of the node in OA counting from A to O, and 
yt, the like for O A’, we have 
1,0 (1 — 2/4) =Jory > Jory <0 
Hence 
1 = iy (@ = Jor 1 7/4a2e* , Jy <1,08; 
seal 2u Tata Ga. nan ee : . (80) 
There will be » roots altogether; but the distribution between the two formule will 
depend on circumstances. In some cases the nodes are all on one side of O. These 
formule lead to some remarkable relations which are true accurately for comple 
rectilinear lakes, and approximately for such as are approximately rectilinear. For 
example, we have a 
(a= ¢:)\a=,) = T/T.) . oa 
In other words, the distance from the ends of the lake of the first node of any pure 
seiche in a given complete rectilinear lake is proportional to the square of the pena of 
the seiche. 
Symmetric TRuNcaTED Lake. 
§ 48. Here a=a’, 8=6’. The equations can be simplified by the suppression of 
unnecessary constants ; and the period equation breaks up into two. We have then — 
we = ALY, (nB)J,(w) — J,(nB)Y(w)} sin n(¢—7), 
= 228 {YB Solve) =F4(nB)Vy(w) \ sin ne) 5 
w/t! = ALY (B)J,(w") — Jy(oB)¥,(w')} sin n(¢ <2), 
fe | Y (8) q(2’) — J, (mB) ¥(w") | sin n(é- =). : . Same 
The period equation is :—for odd oe _ 
Y(f2)Tp(na) — J,(nB)Vy(na) = 5 
Y,(nB)J,(na)-J,(nB)Y,(ma)=0. : . € 3). 
for even nodality, 
SymMMETRIC LAKE SHELVING aT Boru ENDs. 
§ 49. Starting with the formule of §47, we have to put a=a/=p=p'; and 
therefore a=a’. Suppressing unnecessary constants, we may now write 
we = AJ,(w) sin n(t- 7), 
C= MAT (w) sin n(t - 7) ; 
we = AJ, (w’) sin n(t — 7) ; 
Cs - =) Jo(w’) sin n(t — 7) ; : : : (84). 
The period equation breaks up into . = | 
J (va) => 0 ) J ,(no.) =0 fr * . . ° ° (88 ) : 
Hence we have a 
T,=4ra/j, J(gh). . , (86), 
* These formule ire given by Lams in his Hydrodynamics (1895), § 182. 
