H. Sxey.—An Astronomical Telescope on a New Construction. 121 
the surface in the direction PC. Let PC represent this force ; but P is also 
subject to another force, namely, its own weight acting vertically down- 
wards, which we may suppose represented by PQ; the resultant of these, 
therefore, PR, is the whole force acting on P, and so must be perpendicular to 
the surface, and therefore to the curve. To prove that this curve is parabolic— 
NM < MP: PQ: QR (=e FC): 
NM : MPO: W reikt P : Centrifugal force. 
But the dynamical measure of the force of gravity at this latitude is 32-17, 
r 
expressed in feet every second, and of the latter force —-,—(see note), n repre- 
senting 3:1416, or the semi-cireumference of a circle whose radius is 1, ¢ being 
the number of seconds in one revolution, and r bie pening Nh 
A NM 2 Fda se 
D 
4 n? 5 
consequently 3217r + L TEVA fie = = 804 — 4 as N N 
The line NM thus determined is called the sub-normal to the curve at the 
point P, and when the angular velocity of rotation is constant then the sub- 
normal is also constant in length, no matter in what part of the curve the 
point P is situated. This property belongs exclusively to the parabola. 
Hence the surface of a fluid rotating on an axis perpendicular to the horizon 
is a paraboloid. 
To determine then the length of NM for different times of rotation— 
Let ¢ = 1 second then NM=8-04-5 = 0814 feet. 
Pal y p =. 3°08 y 
t=4 s1303 3} 
Now that part of a pòzabölci ahar a ray of light parallel to the axis will be 
reflected along a line forming a right angle to the axis must itself be inclined 
at an angle of 45°, consequently such reflected ray will, when it meets the 
axis, have traversed a distance equal to the length of the subnormal, 
therefore at that part of the curve the two forces, namely gravity and 
` centrifugal force, have the same measure, for they are represented in magni- 
tude and direction by different sides of the same square. 
Moreover this particular ray is the only one which 
will be reflected in a horizontal direction along the 
parameter of the paraboloid until it meets the axis 
in the focus of the curve. And since the distance 
FV equals the half FP’ it also equals half NM, by 
which we can obtain the focal length of the telescope 
for any velocity. 
Within the range of our acquaintance with nature 
we have one remarkable and brilliant metal which 
Q 
