92 REPORT—1858. 
and Ve, / 14g (AH) g (H— —H’), 
or, if g=32, = a/ #8 (aH) M8 (H—H!)= 89-6 (HH). 
This calculation assumes that the sliding is converted into rolling motion in 
an indefinitely short time, as it would in fact be, if the adhesion of the balls 
were large, and the inclination of the planes ¢ small; but if the inclination 
of the latter be considerable, as 15° or upwards, a more exact determination 
is necessary. 
Let, as before, the horizontal components of the velocity with which the 
balls begin to move, be v sin 0, and v cos 0, Z the velocity in the vertical, 
and the inclination of the planes 7 now large. 
The initial velocity of ascent parallel to the planes will be, 
For the ball B........vsin@ cost+Z sin Zz, 
and For the ball By. 2cm sas » cos @ cost+Z sini. 
Let ¢ be the coefficient of frictional adhesion, of the balls to the plane; 
then they will ascend the planes to the heights, 
Bo. nies re ee 
29 2 tani+7o 
B,.tae Bes _(v cos 6 cosi+Z sinz)? 2 tani+5¢ 
2g 2 tani+7¢ 
v and @ are known if the value of Z be given; and this may be ascertained 
experimentally from the compression of the vertical spring; or, as sug- 
gested by my friend Dr. Harte, to whom I have been indebted for these 
equations, a second pair of experimental inclined planes and balls might be 
used, with an inclination greater than 7 (say 22), from the observed movements 
upon which, two more equations could be got, the four equations being then 
more than enough, to determine v, Z and 0. 
But the nature of the instrument is to record the values of H and H,, zn 
terms of the whole time that the balls B and B are out of contact with the 
block gr, @. e. of their rolling up, and down, the inclined planes,—this time 
being given, by the lacune in the pencil-trace made upon the revolving cy- 
linder of paper carried along by the clock. The time of the balls’ ascending 
to the highest point reached on the plane will be independent of adhesion ; 
and calling it ¢, we have, 
For the ball B........ ¢ =? 5inO cost+Z sing 
g sing 
Bor the ball By. \, sage=e eet a 
gsinz 
The time of descent back to the starting-point, due to the heights H and H’, 
will be a little, but inappreciably, less than this. 
The entire time of the double oscillation of each ball, therefore, or its 
movement up and down the plane, as recorded by the instrument, is, 
For B.... T =? Sin. 9 cost+Z sins sin 6 cosi+Z sini (+4/2e 
gsint 2tani+7¢ 
and For B, .. T= oe | 
g sine 2tant+7¢ 
the coefficient } being always =tan a, the angle of sliding for the surface- 
material of the balls upon that of the inclined planes. 
