378 
DR. W. J. MACQUORN RANKINE ON PLANE WATER-LINES. 
n having the series of values 1, 2, 3, See. ; and its radius is given by the formula 
P 
(26 a) 
When there is but one focus, as in the infinitely long curve described in article 10, 
the network of circles is changed into a set of straight lines radiating from the focus, 
and making with AX the series of angles given by the formula 
( 27 ) 
(13.) Component and Resultant Velocities of Gliding. — The component and resultant 
velocities with which the liquid particles glide along the water-lines are given by the 
following equations, in terms of the excentricity a, the parameter f and the coordi- 
nates : — 
udb f[a — x) f(a + x) v_ db__ fy fy 
c dy ' (a—xY + y 12 ' {a + x)‘ 2 -\-y‘ 2 ’ c dx (a— x)* + y*' (a + x)* 1 -\-y 2 ’ j 
u 2 + t> 2 dip db 2 2 f(a—x) 2f(a + x) If* a* f* 
c 2 dy 2 ^~ dx 2 (a — tf) 2 + y 2 ^~ (a + #) 2 + { (a— #) 2 + t/ 2 } . {(a + xY + y 2 ) ’ J 
(28) 
At the point of greatest (breadth that is, at the axis of y) these expressions take the 
following values : — 
! =1 + 
2/a 
=1 + 
P-*_P + t£ m v 0 _ 
= 0 . 
(28 a) 
a?+yl ' cp + yl a 2 + y 2 ’ c 
These equations are applicable to a whole series of water-lines (such as those shown in 
fig. 1), including the generating oval, and are the best suited for solving questions 
relating to such a series. 
But when one particular water-line is in question, it is sometimes more convenient to 
use another set of equations, formed from the equations (28) by the aid of the following 
substitutions, in which 6, as before, denotes - 
-b 
f : 
4a 2 ?/ 2 . 
> 
{(a-xy+y*}.{(a+xf+y*}= sinH 
{(a—xf+tf} + {(a+xf+if} =2a 2 +2x 2 +2y*= 4a 2 +4ay cotan 6 ; 
x 2 -\-y*=a 2 -\-2ay cotan d; x= </{a 2 — y 2 -\-2ay cotan 6). 
These substitutions being made in the equations (28), give the following results 
(29) 
-=1 +- sin s 6 —-- cos*sin«=l+f-^-^ 
c ' a y 1 2a 2a 2 y 
- c =—^sm 2 0= — ^ </{ a 2 —y 2 +2ay cotan 6} sin 2 0; 
-~ 2 = 1 + — sin 2 6— —cos 6 sin 6 sin 2 6 
cay y 
-l+t+n^ (t+^\ 
' a'2y 2 \a'2y <2 J 
f f*\ f 
- 1 ’ cos 2Q — — sin 20. 
y 
( 30 ) 
