An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

149

y' 3spq2 - r (1 +p2)
= Pq 2 — (+ ), where p, q and r are the first, second
7y 2
and third differential coefficients belonging to the point in the
curve.
26. If q, r, s and t be the second, third, fourth and
fifth differential coefficients, it is required to prove that in all
curves of the second order
9q2t -.45qrs + 40r3 =0.
27. Apply Lagrange's Theorem to find a series for v
in terms of nt as far as e3 by means of the equation
(1- e) tan2 =(1 +e) tan2, where t = u - e sin u.
28. If a number a be divided into such a number of parts
x that the continued product of the first, the square of the
second, the cube of the third, &c. shall be the greatest possible,
it is required to prove that log = --.
x?+2a
29. Into how many parts must a given number be
divided, that their continued product may be the greatest possible?
30. Given the three prime factors of a number, required
the indices when the number of its divisors is the greatest
possible.
31. Given the three prime factors of a number, to find the
indices when the sum of its divisors is the greatest possible.
32. If m be an even number, it is required to prove that
27r 4w7
2 - 2 cos- 2 - 2 cos -
m 2 m m
Adn _ + __ + &c
7 _ xm I -- 2 2 r 47qr
1 -2 cos - + x2 1 - 2X cos — + x
m m
to 1m terms. See Article (346).
33. Decompose -- into a series of fractions as in
t t x m +
the last example, both when m is odd and even.