636 PROFESSOR CHRYSTAL 
If we transform (71) by putting w=2na,/(1—a/a)/,/(gh), and P = Rw, it becomes 
ote lipo, . ll 
dw? aw dw 
which is a particular case of the Besse, Equation.* 
If J,(w) and Y,(w) denote the BesseL and Neumann Functions, as defined in Gray 
and MaTHEws’ treatise, the general solution of (72) is 
R=AJ,(w)+B Y,(w) . 
Hence, with a slight adaptation of the constants, we find 
&w={AJ,(w) + BY,(w)} sin x(t- 7); . , (73) ; 
2ZajAd Bd : 
C= + | eal that) + 2 (w¥y()) } sin n(t—T). 
Now, by one of the fundamental properties of J,(w) and Y,(w), we have 
1 d 1 Ad , : é x 
= wJ,(w) ) oa J q(t), At ale 1) ) = Y,(w) : 
Hence 
ee J,(w)+BY,(w)} sin n(t—r) : ; ‘ ; : (74). 
From (73) and (74) solutions for the following particular cases are readily obtained. 
RECTILINEAR LAKE witH Two SLOPES TRUNCATED AT BotH ENDs. 
S46. The laws of depth for the two parts will be given by A(1—a/a) and 
h(1+«/a’), if we take the origin at the junction of the two slopes, and choose the 
standard case to be that where the bottom slopes upwards on both sides of the junction. 
NY p ) Dp A 
Fie. 8, 
Let 
w= 2na,/(1 — x/a)/ /(gh), w' =2na’ /(1+2/a’)//(gh) . ; ; ‘ (75) ; 
and ; 
a= 2a/J/(gh), P= 2an/(1 — p/a)//(9h) ; 
a =2a'//(gh), B= 2a /(1 -p'/a’)// (gh). 
* Readers unacquainted with the properties of Brsspi Functions will find all that is here required in a few 
pages/of the treatise by Gray and Marumws (1895), ch. ii. and pp. 241-292 containing the tables, 
