260 THE EQUILIBRIUM OF KEVOLVING LIQUID [387
In order to trace the various curves we may calculate by quadratures from (4) the position of a sufficient number of points. This, as I understand, was the procedure adopted by Beer. An alternative method is to trace the curves by direct use of the radius of curvature at the point arrived at. Starting from (7) we find
n
ds* ~ \ a? a ds ' and thence
a
n ^
(18)
- .,
p dx/ds~ a2
From (18) we see at once that fi = 0 makes p = a throughout, and that when O = 1, x = 0 makes p = oo .
In tracing a curve we start from the point A in a known direction and with p = a/(2fl + 1), and at every point arrived at we know with what curvature to proceed. If, as has been assumed, the curve meets the axis, it must do so at right angles, and a solution is then obtained.
The method is readily applied to the case £1 = 1 with the advantage that we know where the curve should meet the axis of y. From. (18) with H = 1 and a = 5,
(19) ( y)
1000
Starting from x = 5 we draw small portions of the curve corresponding to decrements of x equal to %2; thus arriving in succession at the points for which x = 4*8, 4'6, 4'4, &c. For these portions we employ the mean curvatures, corresponding to x — 4'9, 4'7, &c. calculated from (19). It is convenient to use squared paper and fair results may be obtained with the ordinary ruler and compasses. There is no need actually to draw the normals. But for such work the procedure recommended by Boys* offers great advantages. The ruler and compasses are replaced by a straight scale divided upon, a strip of semi-transparent celluloid. At one point on the scale a fine pencil point protrudes through a small hole and describes the diminutive circular arc. Another point of the scale at the required distance occupies the centre of the circle and is held temporarily at rest with the aid of a small brass tripod standing on sharp needle points. After each step the celluloid is held firmly to the paper and the tripod is moved to the point of the scale required to give the next value of the curvature. The ordinates of the curve so drawn are given in the second and fifth columns of the annexed table. It will be seen that from x — 0 to x = 2 the curve is very flat. Fig. (1).
* Phil. Mag. Vol. xxxvi. p. 75 (1893). I am much indebted to Mr Boys for the loan of suitable instruments. The use is easy after a little practice.cur with small spheres as with large ones. The most interesting case occurs when one sphere is very large relatively to the other, as when a grain of sand impinges upon a glass surface. If the velocity of impact be given, the glass is as likely to be broken by a small grain as by a much larger one. It may be remarked that this conclusion would be upset if rupture depends upon the duration of a strain as well as upon its magnitude.