42 THE EFFICIENCY OF PADDLE WHEELS WHEN DEEPLY IMMERSED. 
equal velocity from the two ends of the wire, 
and occurs latest in the middle of the circuit 
3d, The light of electricity in a state of high 
tension, has a less duration than the millionth 
part of a second. 4th, 'I’he eye is capable of 
perceiving objects distinctly which are pre- 
sented to it during the same small interval 
of time. 
PHYSICS, &c. 
Mr. Hamilton’s paper on a general me- 
thod in Dynamics is a most elaborate 
one. He shews that in the method for- 
merly employed to develope the laws of 
motion, the determination of the motion of a 
free point in space, depends on the integra- 
tion of three equations, in the ordinary differ- 
entials of the second order, and the determi- 
nation- of the motions of a system of free 
points attracting or repelling one another, 
depends on the integration of a system of 
such equations, in number threefold the 
number of the attracting or repelling points, 
unless we previously diminish by unity this 
latter number, by considering only relative 
motions. Mr. Hamilton’s method is to reduce 
the problem to the search and differentiation 
of a single function, which satisfies two 
partial differential equations of the first order, 
and of the second degree, and every other 
dynamical problem respecting the motions of 
any system, however r?umerous, is reduced, in 
like manner, to the study of one central 
function. 
THEORY OF CLAIRAUT. 
IMr. Ivory demonstrates that the beautiful 
theoiy of Clairaut, which assumes for the 
foundation of its superstructure, a mass of 
fluid in equilibrium, and that the pressure of 
every new stratum upon the surface of which 
it is laid, is caused solely by the forces in 
action at that surface, is very satisfactory 
when no cause of motion emanates from the 
fluid itself, and all theforces in action depend 
merely on the place of a particle, but is de- 
fective w'heu applied to fluids consisting of 
particles that act upon one another by attrac- 
tion or repulsion Clairaut having omitted to 
attend to the attraction of the stratum, which 
is not infinitely little in its effect upon the 
motion of a particle, and is expressed by the 
difference of two definite integrals. The 
correction of Clairaut’s theory is very impor- 
tant ; because, to him belongs an essential 
part of the theory of the earth, and he was the 
first that entertained correct notions respect- 
ing the effect to alter the form of the terra- 
queous globe, produced by heterogeneity in 
its structure. In the theory of the French 
philosopher, tire equations of the upper sur- 
face of the fluid, and of all the level surfaces 
underneath it, are derived from the single 
expression of the hydrostatic pressure, and are 
dependant on the differential equation of the 
surface. 
They require, therefore, that this latter 
equation be determinate and explicitly given ; 
and accordingly, they are sutficient to solve 
the problem, where the forces are known al- 
gebraical expressions of the co-ordinalcs of 
the points of action, but they are not suffici- 
'ent when the forces are not explicitly given, 
but depend as they do in the case of a homo- 
geneous planet on the assumed figure of the 
fluid. In the latter case, the solution of the 
problem requires farther, that the equations 
be brought to a determinate for,r), by elimi- 
nating all that varieswith the unknown figure 
of the fluid. 
The author establishes a theory on the 
subject, applies it to the principal problems 
ofthe equilibrium of a homogeneous fluid at 
liberty, and demonstrates that the figure of 
equilibrium of a homogeneous planet can be 
no other than an oblate elliptical spheroid of 
revolution. 
ON THE EFFICIENCY OF PADDLE 
WHEELS. 
Mr. Barlow draws the following inferences 
from the results of various experiments made 
to determine the efficiency of paddle-wheels 
of steam-boats, so constructed as to make 
the floats enter and leave the water nearly in 
a vertical position, as compared with common 
wheels, and with relation to the consumption 
of fuel ; 1. When the wheel is but slightly 
immersed, little advantage is gained by the 
vertically acting paddle; 2. When deeply 
immersed, the vertical paddle has considera- 
ble advantage over the common wheel. 
3. When the position of the common wheel 
is vertical, it affords less resistance to the 
engine, and is less effective than in any part 
of its revolution, which is exactly reversed 
in the case of the new wheel. 
4. In any wheel, the larger the paddles 
the less is the loss of power; because, the 
velocity of the wheel is not required to 
exceed that of the vessel in so high a degree, 
in order to acquire the resistance necessary to 
propel the vessel. 
5. With the same boat and the same wheel 
no advantage is gained by reducing the pad- 
dle so as to bring out, as it is called, the full 
power of the engine, the effect produced 
being merely to increase the speed of the 
wheel, and consume steam to no purpose. 
6. With the same boat and the same wheel 
the speed will be increased by diminishing 
the diameter, or by reefing the paddles, the in- 
crease of speed being in the ratio of the square 
roots of the radii, or the cube roots of the pow- 
ers employed. This is important in long voy- 
ages, where the immersion of paddles is great, 
in consequence of the quantity of the coals 
with which steam vessels are required to be 
laden. An increase of speed will be given, 
amounting to nearly one mile per hour, by 
reducing the diameter of the wheel so as to 
allow the engine to perform its full duty. 
7. An advantage would be gained by a 
wheel of large diameter, as far as the immer- 
sion of the paddle produced by loading the 
vessel is concerned, as it would not so sen- 
sibly affect the angle of inclination at which 
it euiered the water. But, to have large 
wheels, it is necessary either to have the 
engines made with long strokes, or to have 
the paddle-wheel on a different shaft, in order 
to diminish the speed. 
