[644
645]
39
645.
A SMITH’S PRIZE PAPER, 1877.
). 168.]
[From the Messenger of Mathematics, voi. vi. (1877), pp. 173—182.]
The paper was as follows:
die ordinary process,
which occupies any
^onals (N.W. to S.E.)
2, 3,.., 20, 21, 20',
impartments respect-
on the top line be
ise may be) denotes
1. Show (independently of the theory of roots) how, if x satisfies an equation
of the order n, a given rational function of x can in general be expressed as a
rational and integral function of the order n — 1. State the theorem in a more
precise form, so as to make it true universally.
2. Investigate the form of the factors of 1 + sin (2n +1) x considered as a
function of sin x ; and give the formulae in the two cases, 2n + 1 = 3 and 5 respectively.
3. Write down the substitutions which do not alter the function ab + cd; and
explain the constitution of the group.
4. Find in a form adapted for calculation an approximate value for the sum of
the middle 2a +1 terms of the expansion of (1 + Yf n , n being a large number, and
a small in comparison therewith.
be retained only if
[•miilse for a square
ilse for the general
Obtain thence a complete and precise statement of the theorem that in a large
number of tosses the numbers of heads and tails will probably be nearly equal.
5. A point in space is represented on a given plane by its projections from
two fixed, points. Show how a problem relating to points, lines, and planes, is
thereby reduced to a problem in piano ; and apply the method to construct the line
of intersection of two planes each passing through three given points.
6. A weight is supported on a tripod of three unequal legs resting on a smooth
horizontal plane, their feet connected in pairs by strings of given lengths. Show how
to determine the tensions of the several strings.
7. Explain the ordinary configuration of a system of isoparametric lines on a
spherical surface ; for instance, what is the configuration when there are two points
of minimum value, and one point of maximum value, of the parameter?
