L80 REPORT—1863. 
Each of these springs will fit every instrument alike. All of the scales except 
No. 2 are multiples of 8; and the common rule will measure all the diagrams, if the 
proper scale is not at hand. It will be observed that the five higher scales do not 
indicate the vacuum, as the greater number of engines which work steam at high 
pressures do not condense, and, moreover, at these pressures the scale of the indi- 
cation necessarily becomes small, while it is always highly desirable to show the 
vacuum on a large scale. Spring No. 1 may be employed to indicate the vacuum 
in engines which work steam at high pressures and with condensation. It can be 
readily substituted in the indicator, and the diagram which it will give will be on 
a satisfactory scale. It is provided with a stop, which prevents it from being com- 
pressed too much, so that a high pressure of steam will not injure it. Moreover, 
the vacuum being omitted from the scales which go above 60 lbs., the entire range of 
the pencil is available for the pressures above the atmosphere, which are therefore 
shown on a somewhat larger scale. The springs indicating pressures above 60 lbs. 
will be made, however, to indicate the vacuum also, when so ordered. 
The springs are tested with a highly sensitive gi Poe expressly designed for 
the purpose, and are corrected for a temperature of 212°, which is the temperature 
at which they will work under almost all circumstances, and at which their accu- 
racy is guaranteed. 
Thompson’s Universal Stopper for Bottles, §&c. By D. Pusrrer. 
An Investigation on Plane Water-lines. 
By W. J. Macquorn Ranxrne, C.Z., DL.D., P.R.SS.L. & E., &e. 
1. This paper contains an abstract of a mathematical investigation which has 
been communicated in detail to the Royal Society (see Phil. Trans. 1864). By 
the term “plane water-line” is meant one of those curves which a particle of 
a liquid describes in flowing past a solid body, when such flow takes place in plane 
layers. Such curves are suitable for the water-lines of a ship; for during the mo- 
tion of a well-formed ship, the vertical displacements of the particles of water near 
the surface are small compared with the dimensions of the ship *. 
2. The author refers to the researches of Professor Stokes (Camb. Trans. 1842) 
“On the Steady Motion of an Incompressible Fluid,” and of Professor William 
Thomson (made in 1858, but not yet published), as containing the demonstration 
of the general principles of the flow of a liquid past a solid body +. 
3. Every figure of a solid, past which a liquid is capable of flowing smoothly, 
generates an endless series of water-lines which become sharper in their forms as 
they are more distant from the primitive water-line of the solid. The only 
exact water-lines whose forms have hitherto been completely investigated are 
those generated by the cylinder in two dimensions, and by the sphere in three 
dimensions. In addition to what is already known of these lines, the author points 
out that when a cylinder moves through still water, the orbit of each particle of 
water is one loop of an elastic curve. 
4, The profiles of waves have been used with success in practice as water- 
lines for ships, first by Mr. Scott Russell (for the explanation of whose system the 
author refers to the ‘Transactions of the Institution of Naval Architects’ for 
1860-62), and afterwards by others. As to the frictional resistance of vessels 
having such lines, the author refers to his own papers—one read to the British 
Association in 1861 and printed in various engineering journals, and another read 
to the Royal Society in 1862 and printed in the Philosophical Transactions t. 
5. The author proceeds to investigate and explain the properties of a class of 
water-lines comprising an endless variety of forms and proportions. In each series 
* As water-line curves have at present no single word to designate them in mathema- 
tical language, it is proposed to call them Neotds, from vnis, the Ionic genitive of vais. 
t See also a paper by Dr. Hoppe, in the ‘ Quarterly Journal of Mathematics’ for 
March 1856. 
} See also a paper by the author on the Computation of the Probable Engine-power and 
Speed of proposed Ships, in the Transactions of the Institution of Naval Architects for 1864: 
also a treatise on “ Ship-building,” 1864. 
