284 
PROFESSOR W. J. MACQUORN RANKINE ON THE 
distances of the point (x, y) from the four foci respectively. The equation of the cycnoid 
of revolution which produces the series of stream-lines is\|/=6 = 0; and this equation 
has two roots, viz. y= 0, denoting the axis of x, and 
y-—k 2 (cos0 2 — cos0j)+/i’' 2 (cos0 4 — cos 4,), (42 a) 
in which 0 2 , 0 3 , and 0 4 denote the angles made with the axis of x by lines drawn from 
the point (x, y) to the four foci. 
The component comparative velocities are as follows : 
U ydy 
-x+a x + a 
k' 2 / — x -f a' 
x + 
a ~y, 
djf k 2 y ( l , 1\ . k n y I 1 1 \ 
1 ydx 2 y 2 y r l' r 4 /’ 
(43) 
When the two ends of the solid are symmetrical, we have a"=a ' ; and the value of u at 
the midship section, where v=0 and #=0, is as follows, 
«o=l 
k 2 a 
(« 2 + */o) f 
k' 2 a' 
(a n +y 0 /’ 
(43 a) 
in which y 0 is the midship half-breadth. 
§ 11. Cycnoids of Revolution. — Extreme Dimensions . — Let l denote the distance from 
the origin of one of the points where the cycnoid surface of revolution cuts the axis of x. 
Then, in the first of the equations (43), makingw=:0, y=0, x=l, we obtain the following 
equation of the eighth order, 
0 = (l 2 -a 2 ) 2 . (l—a!) 2 (l-\-a!') 2 — e lk 2 la(l— a!) 2 (I « ,; ) 2 1 
-y|2 l(a" + a')+(a" 2 -a'*)} • (l 2 -a 2 ) 2 . 
(44) 
The greatest positive and greatest negative real roots of this equation give the ends of 
the cycnoid ; the other real roots belong to internal stream-lines. 
When the ends of the solid are symmetrical, so that a"=a, the preceding equation 
becomes 
0=(l 2 -aJ(l 2 -a!J-2k 2 la(l 2 -a' 2 y-2E 2 la!(l 2 -a 2 Y (44 a) 
The greatest half-breadth and its position are to be found in the general case, as before, 
by deducing values of y and x by elimination from the pair of equations 4/ = 0, v=0. 
When the ends of the solid are symmetrical, the greatest half-breadth is at the origin ; 
hence, making x — 0, v—0, we have the following equation, 
2 2 k 2 a 2 k ,2 a! 
~y°— -v/(a 2 + 2$— V(«' 2 + ?/ 2 ) ’ 
(45) 
which, when reduced to the form of an algebraic equation with y\ for the unknown 
quantity, is of the eighth order, as follows : 
0 =yl{yl + « 2 )%o + «' 2 ) 2 + 1 §k*a\y\ + a! 2 ) 2 ' 
+ 16^V 4 (y 2 +a 2 ) 2 — 8^V^(^+a 2 )(yo + «' 2 ) 2 x 
(45 a) 
-8/ WyKyl+aJiyl+a' 2 ) 
+ 1 Qk 4 k u a 2 a' 2 (y 0 -f- a 2 )(yl -f- a! 2 ). 
