390 
DE. W. J. MACQTTOEN EANKINE ON PLANE WATEE-LINES. 
(30.) Provisional Formula for Resistance . — Until the difficulty of integration, men- 
tioned in article 30, shall have been overcome, or until more exact experimental data 
than we have at present shall have been obtained, the following provisional formula, 
analogous to that which has been found to agree with the results of experiment on 
trochoidal and nearly trochoidal lines, as well as some others, may be considered as a 
probable approximation for lissoneoids, 
K=^.(i+4<^)lG; (53) 
where G is the mean girth of the vessel under water ; L her total length ; u 0 the mid- 
ship velocity of gliding, found, for a lissoneoid, by equation (37) of article 17 ; c the 
speed of the ship; W the heaviness of water; and K a coefficient of friction (= about 
•0036 for a clean surface of paint). 
Appendix. 
Note to Article 11. — The general process of constructing a series of curves whose 
equation is <p(#, y)-\-\p(tv, y) = constant, by drawing lines diagonally through a network 
consisting of two sets of curves whose equations are respectively <p(tc, y) — constant and 
y) = constant, is due to Professor Clerk Maxwell. 
Art. (1.) 
( 2 .) 
(3.) 
( 4 .) 
( 5 .) 
Art. (6.) 
(?•) 
( 8 .) 
. ( 9 0 
( 10 .) 
( 11 .) 
( 12 .) 
(130 
(14.) 
Summary op the Contents. 
Section I. — Introduction , and Summary of known Principles. 
Plane Water-Lines in two Dimensions defined. 
General Principles of the Flow of a Liquid past a Solid. 
Notation. 
General Characteristics of Water-Line Functions. 
Water-Line Curves generated by a Circle, or Cyclogenous Neoids. 
Section II . — Properties of Water-Line Curves generated from Ovals , 
or Oogenous Neoids. 
Derivation of other Water-Line Curves from Cyclogenous Neoids. 
General Equation of Oogenous Neoids. 
Geometrical Meaning of that Equation. 
Properties of Primitive Oval Neoids. 
Varieties of Oval Neoids, and extreme cases. 
Graphic Construction of Oval and Oogenous Neoids. 
Graphic Construction of Cyclogenous and Parabologenous Neoids. 
Component and Resultant Velocities of Gliding. 
Trajectories of Normal Displacement, and of Swiftest and Slowest Gliding. 
