; 
ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 665 
function have also been computed directly from the series by Professor CHRysTAL, assisted 
by Dr Bourcexss, Professor Gipson, and Mr Horssurcu. They were found to be 2°76 
and 12°34, and hence are remarkably close to the values given in (35). The root 28°23 
was also verified by Dr Burexss, Professor Gipson, and myself. I have convinced 
myself that the exact value must certainly lie between 28°229 and 28°230, so that the 
second decimal place of the interpolated value appears to be correct. The method of 
interpolation here adopted places us therefore in a position to determine the infinite 
number of roots of the two transcendental equations ©_,(¢, 1)=0, and S_,(c, 1)=0, 
~ and thus enables us to determine the periods of the seiches in lakes with convex parabolic 
floors from Professor CHrysTAL’s equation :—T,=27a/ Jel (see Proc. RS.E., vol. xxv. 
p- 332). 
A similar method of interpolation may be employed in the calculation of the position 
of the nodes. These are found by first forming the equations 
d@_(w) 6 AG, (w) B) IG, (w) a AG, (w) 6G 
[SS , Saye ? dw J es — | ? 
dw du dw 
AS_,(w) _ 0 dSo(w) _ 9 AS (10) _ 0 AS,(w) _ 0 (36) 
dw ; dw ‘ dw ‘ dw y 
and by determining the values of w satisfying these equations under the condition that 
¢, has the values mentioned in (85). By interpolating between the w thus obtained, 
we find with sufficient approximation the successive values which satisfy the two Seiche- 
equations 
d@_,(10) =Q0 and dS_,(w) =0 
dw dw 
? 
and which therefore fulfil the required condition that the vertical displacement ¢= 0. 
In this way I have found the following positions of the nodes in convex parabolic lakes : 
Uninodal Seiche w= 0 
Binodal Seiche w= +0°473 .... 
Trinodal Seiche w=0; +0°632.... 
Quadrinodal Seiche w= +0°224....; t0°717.... 
It is interesting now to investigate the positions of the nodes under the various 
lake. From Professor Curystat’s investigations and the preceding discussion we find for 
| Lake with poe ao . Trinodal Seiche. — (uadrinodal Seiche. 
¢ wR . . 
“coneave parabolic floor, . .| w=0 + 577 0; +775 | + -340; + -862 
plain horizontal ‘ 0 +500 | 0; + °667 | ae 200 5 “750 
‘convex parabolic _,, 0 Ane | dOuved-' 632 QA 4 717 (37) 
“Convex quartic, 0 +447 | 0; + :600 + 202; + -684 
t | 
The figures show clearly that in lakes with curved floors the nodes are always displaced 
) towards the shallow water. 
