DR. W. J. MACQLORN RANKINE ON PLANE WATER-LINES. 
385 
In preparing these formulae for integration, it is necessary to express the function to 
be integrated in terms of constants and of the independent variable only, x, y, 0, or q, 
as the case may be ; for example, if y or 6 is the independent variable, the expression 
of the function to be integrated is to be taken from the equations (30) of article 13. 
Owing to the great complexity of that function, its exact integration presents diffi- 
culties which have not yet been overcome, although a probable approximate formula 
for the resistance has been arrived at by methods partly theoretical and partly empirical, 
as to which some further remarks will be made in the third section of this paper*. 
There is one particular case only in which the exact integration of equation (46 a) is 
easy, that of a complete circular water-line of the radius l ; and the result is as follows : — 
.......... (48) 
(22.) Statement of the General Problem of the Water-Line of least Friction. — It is 
evident that, by introducing under the sign of integration in equation (18) of article 7 
an arbitrary function of x', the integral may be made capable of representing an arbi- 
trary function of x and y, and will still satisfy the condition of perfect liquidity; and 
thus the equation 
i= y + )-.¥=w+f =0 (48a) 
may be made to represent an arbitrary form of primitive water-line. 
To find therefore, by the calculus of variations, the water-line enclosing a given area 
which shall have the least friction, will require the solution of the following problem : — 
To determine the function <p(x') so that, with a fixed value of the integral jxdy, the inte- 
gral in equation (46 a) shall be a minimum. 
(22 a.) Another Class of Plane Water-Line Equations. — A mode of expressing the 
conditions of the flow of water in plane layers past a solid, differing in form from that 
made use of in the preceding parts of this paper, consists in taking for independent 
variables, not the coordinates of the water-lines themselves, x and y, but the coordi- 
nates of their asymptotes ( b ), and of the asymptotes of their orthogonal trajectories (q). 
These new variables are connected with x and y, and with the velocity of gliding, by 
the following equations : — 
id+v 2 dq db dq db 
c 2 dx ’ dy dy * dx' 
‘dx dy 
dq * db 
dy dx 
dq db 
It can be shown that in order to satisfy the condition of liquidity we must have 
y- Tl ’ 
db 
(49) 
(50) 
* See the Civil Engineer and Architect’s Journal for October 1861, the Philosophical Transactions for 1863, 
the Transactions of the Institution of Naval Architects for 1864, and a Treatise on Shipbuilding, published in 
1864. 
3 p 2 
