40 
a smith’s prize paper, 1877. 
[645 
8. Find the attraction of an infinite circular cylinder, of uniform density, on a 
given exterior or interior point. 
9. Determine the number of arbitrary constants contained in the equation of a 
surface of the order r which passes through the curve of intersection of two given 
surfaces of the orders m and n respectively. 
10. Find, for the several values of p, the number of the conics passing through 
p given points and touching 5 —p given lines; and, in each case, show how to obtain 
(in point-coordinates or line-coordinates, as may be most simple) the equations of the 
conics satisfying the conditions in question. 
11. Investigate the theory of the linear transformation of a ternary quadric 
function into itself. 
12. Explain the theory of the solution of a partial differential equation, given 
function of x, y, z, p, q, r — arbitrary constant H; where p, q, r are the differential 
coefficients of the dependent variable u in regard to the independent variables x, y, z 
respectively. 
I propose, not (as in former years) to give complete solutions, but only to notice 
in more or less detail the leading points in the several questions. 
1. The expression is of course required, not only for a given integral function 
of x, but for a given fractional function. The case where the given function is 
integral presents no difficulty; when the given function is fractional, the most simple 
case is where it is = ^ - ■; supposing the equation to be f (x) = 0, here dividing 
f(x) by x — a, we have a quotient R (x) which is a rational and integral function of 
an order not exceeding n — 1, and a remainder which is = f(a); that is, 
/0) 
x — a 
= R (x) + 
/0). 
x — a 
or, in virtue of the given equation ——^ = — R (x), viz. we have thus —-— 
x — a v x — a 
required form. But if f(a) = 0, then we do not obtain such an expression of 
in the 
It has to be shown that the like considerations apply to any fractional function, and 
the precise form of the theorem is, that any rational function of x which does not 
become infinite for any value of x satisfying the given equation, can be expressed as a 
rational and integral function of an order not exceeding n — 1. 
2. The function 1 — sin (2n + 1) x is a rational and integral function of sin x, of 
the order 2n+l; which if n is even (or 2?i + 1 = 4p + 1) contains, as is at once 
seen, the factor 1 — sin x, but if n is odd (or In + 1 = 4<p — 1) the factor (1 -f sin x). 
Suppose that any other factor is 1 — , where sin a not = ± 1; then this will be 
a double factor if only sin x — sin a satisfies the condition 
d. sin x 
{1 — sin (2n + 1) x\,