MATHEMATICAL THEORY OF STREAM-LINES . 
293 
tion of the surface of the liquid at an, indefinite distance from the solid above the centre 
of mass of the liquid ; so that the disturbance of head at any point relatively to that, 
centre of mass is expressed as follows : — 
h — h 0 -\- 
V 2 E 
-7/b 
(64) 
The last term vanishes when the volume of liquid L increases indefinitely. 
When the trace of a disturbing solid, together with its external stream-lines and lines 
of disturbance, has been drawn, as in figs. 2 and 3, the manner in which the disturbances 
of motion and of head vary at different points may be represented to the eye by means 
of a diagram like fig. 5, constructed as follows. Draw a straight line A 13 to represent 
the velocity of the undisturbed current (equal and opposite to the velocity of the ship). 
From A draw a series of straight lines, such as A C, A C', A C", parallel to a series of 
tangents at a series of points in the trace of the solid. From B draw a series of straight 
lines, such as B C, B C', B C", parallel to the tangents of the lines of disturbance at the 
same series of points, cutting the first-mentioned series of lines in C, C, C", Then in 
each of the triangles in the diagram, such as A B C, corresponding to a given point in the 
trace of the solid, B C will represent the direction and velocity of the disturbance, A C 
the direction and velocity of the elementary stream of liquid relatively to the solid ; and 
AB 2 — AC 2 
the disturbance of head, positive upwards, will be expressed by ^ 
At the points marked L and L' in figs. 2, 3, and 4, the disturbance of head is simply 
the height due to the velocity of the disturbing solid. 
When the disturbances of head, as in a liquid with a free upper surface, take the form 
of disturbances of level, they produce two effects — alteration of the forms and motion of 
the elementary streams, and the formation of waves ; which waves may give rise to a 
particular kind of resistance. In the present paper it is assumed that the dimensions of 
the disturbing solid are so large, or its motion so slow, that the effects of the disturbances 
of level on the forms and motions of the elementary streams may be neglected ; and the 
investigation in the ensuing sections is confined to the action of those disturbances in 
producing waves and wave-resistance. 
§ 17. Virtual Depth and Speed of Waves. — The term virtual depth of longitudinal 
disturbance , or, more briefly, virtual depth, is used to denote the depth found by inte- 
grating the velocity of longitudinal disturbance throughout a vertical column of a liquid 
mass, and dividing the integral by the value of that velocity at the free upper surface of 
the mass. For example, let u— 1 be the velocity of longitudinal disturbance in a given 
indefinitely slender vertical column at the depth z, and u x — 1 its value at the surface ; 
and let Z be the virtual depth ; then 
r/ ^ {u — 1 ) dz ' 
* A ? 
Mj — 1 
( 65 ) 
