DR. J. W. MACQUORN RANKINE ON PLANE WATER-LINES. 375 
the condition stated in article 4, equations (13). The other solution represents the 
oval LB. 
The greater semiaxis of that oval, OL, will be called the base of the series of water- 
lines generated by the oval, and denoted by l ; its value is found as follows : 
db . »d / , y , . y \ a—x , a + x ) 
= 1 +/^( tan + tan ^- x ) =1 +f\(a-xr+f + J^V)>+j*\ ; 
but at the point L we have 
and therefore 
whence 
*=*;*= °i fy=°- 
0=1+ /(*L+«v?) ; 
l 2 =a?-\-2af. . . . 
( 21 ) 
To find the parameter f when the base l and excentricity a are given, we have the 
formula 
P-a* 
2 a ’ 
( 2 - 
The half-breadth, or minor semiaxis of the oval, OB =y 0 , is the root of the following 
transcendental equation, found by making #=0 in equation (19), 
y.-2/tan-£=0, (23) 
which may be otherwise written as follows : — 
tan|-^=0 (23 a) 
When the minor semiaxis y 0 and excentricity a are given, the parameter /’is found by 
the equation 
f=— Vi — a -, (24) 
2 tan Vo 
and thence the base l can be computed by equation (21). 
When the base l and half-breadth y 0 are given, the excentricity a is found by solving 
the following transcendental equation : — 
°yo— (F— « 2 )tan- 1 ^-=0 (24 a) 
Vo 
An oval neoid differs from an ellipse in being fuller towards the ends and flatter at 
the sides ; and that difference is greater the more elongated the oval is. 
(10.) Varieties of Oval Neo'ids, and extreme cases. — The excentricity a may have any 
value, from nothing to infinity; and the base l may bear to the half-breadth y Q any 
proportion, from equality to infinity. When the excentricity a= 0, the two foci coalesce 
with the centre O ; the base l becomes equal to the half-breadth b ; the oval becomes a 
mdccclxiv. 3 e 
