274 
PEOFESSOE W. J. MACQTJOEN EANKINE ON THE 
are the traces upon that surface of the several surfaces expressed by = will be 
represented by lines on that piece of paper ; and each of those lines will have an asym- 
ptote, being the trace, on the surface %=<?, of a surface whose equation is ^ 0 =b. 
The drawing of such stream-lines is facilitated by the following process invented by 
Mr. Clerk Maxwell : — when a function ^ is the sum of two more simple functions, 
'4'o+'4'n draw the series of lines whose equations are ^ 0 =b 0 ; then draw the series of 
lines whose equations are i ^ l = b l ; then draw curves diagonally through the angles of 
the network made by the two former series of curves, in such a manner that at each 
intersection shall be =b ; the new series of curves will be that represented by 
the equation \p 0 -{-'<p l = b. The same process may be extended to curves represented by 
a function consisting of any number of terms. For example, let the function be one of 
three terms, + ^ 2 - Draw the two series of lines represented respectively by 
\Jy 2 =b. 2 and \P l =b 1 ; through the angles of the network draw the series of lines repre- 
sented by + J then draw a fourth set of lines, being those represented by 
4, 0 —b 0 , and through the angles of the network made by the third and fourth series of 
lines, draw a fifth series of lines, being that represented by 
4 / o^r4 l i~ > r4 l 2 = bo-\-b l -\-b 2 =b. 
Figs. 2 and 3 show examples of those processes ; and in fig. 4 also the curves have been 
drawn by means of them, although the network is omitted. 
In each case the lines expressed by the function \|/ 0 represent a uniform current; 
and in the figures they are straight and parallel to x. The lines expressed by ^ — 
the sum of the remaining terms of the function, which form a network with the lines 
of uniform current, may be called Lines of Disturbance ; for each of them indicates the 
direction of the motion of disturbance of each particle that it traverses. They are 
marked with bold dots. 
§ 4 a. Empirical Mule as to the volume enclosed by a Stream-line Surface. — It has 
been found by the drawing and measurement of a variety of figures bounded by closed 
stream-line surfaces, unifocal, bifocal, and quadrifocal, and also by parts of bifocal stream- 
line surfaces suited for the shapes of vessels, that the following rule gives the volume 
contained within such a surface to the accuracy of about tw r o per cent. : — multiply the 
area of midship (or greatest transverse) section by five sixths of the longitudinal distance 
between the pair of transverse sections whose areas are each equal to one third of the 
area of midship sectionf . 
* Note added in June 1871. — The values of l are supposed to be equidifferent. 
f This rule was first published as applied to stream-lines in two dimensions, in a treatise entitled ‘ Ship- 
building, Theoretical and Practical,’ bj Watts, Eankine, Napier, and Barnes : Glasgow, 1866, page 107- Its 
approximate correctness extends to such extreme cases as a sphere on the one hand and a wave-line bow on the 
other. 
