PAPER ; SOLUTIONS. 
487 
534] 
SMITHS PRIZE 
the equations are 
dH _ 
dr 
dH _ 
dv 
dH _ 
dy 
dv 
dt’ 
dy 
dt’ 
dy ~ 
dt’ 
dH _ 
dv 
dH _ 
dy 
dH _ 
_dy 
dr 
~dt’ 
dv 
~dt’ 
dy ~ 
dt ' 
We have 
dH_ dH_ v dH_y 
dr ’ dv r 1 2 cos' 2 y ’ dy f 2 ’ 
dH _ 1 / v 2 
dr r 3 vcos 2 y ^ 
d V dH _ dH _ v 2 sin y 
dr ’ dv ’ dy r 2 cos?y ’ 
and, substituting these values, the equations of motion present themselves as six equations 
of the first order between r, v, y, r, v, y, and t in the form 
( lt — dr_ dv _ dy 
r v y 
r 2 cos 2 y r 2 
dv 
dv 
1 / v 2 A dV 0 
r 3 Vcos 2 y y J dr 
dy 
v 2 sin 2 y * 
r 2 cos 3 y 
12. An unclosed polygon of (m +1) vertices is constructed as follows: viz. the 
abscisses of the several vertices are 0, 1, 2 ... rn, and, corresponding to the abscissa k, the 
ordinate is equal to the chance of (to + k) heads in 2to tosses of a coin; and to then 
continually increases up to any very large value: what information in regard to the 
successive polygons, and to the areas of any portions thereof, is afforded by the general 
results of the Theory of Probabilities ? 
It is somewhat more convenient to take account also of the abscissse —1, —2,...,— to, 
thereby obtaining a polygon of 2to +1 vertices, symmetrical in regard to the axis of y. 
In such a polygon, the sum of the 2to +1 ordinates is = 1 ; the central ordinate is 
the largest, and the ordinates continually diminish as k increases: moreover for any 
large value of to the area of the whole polygon is very nearly, and may be regarded 
as being, = 1; and the area between the ordinates corresponding to the abscissae + k, 
— k as being equal to the probability of a number of heads between m + k, to — k, in 
the 2to tosses of the coin. A general result of the Theory of Probabilities is that in 
a great number of trials the several events tend to happen in the proportion of their 
respective probabilities; viz. in the case of the 2m tosses there is a tendency to an 
equal number of heads and tails. But observe that this does not mean that the 
probability of to heads and to tails increases with the number 2to of the trials; or 
even that, a being any given number, the probability of a number of heads between 
to + a and to — a increases with the number 2to of trials; on the contrary, it diminishes; 
what it does mean is that taking the limit of deviation to vary with to, say a number 
of heads between to + am, m — am, the probability of such a number increases with to ; 
viz. that taking a a fraction however small, to can be taken so large that the 
probability of a number of -heads between to + am, m — am in the 2to trials, shall be as 
nearly as we please = 1.