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

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67. From the same equation determine the values of the
functions y-m, log y, sin y, el y, log (2y ) and sin (1-.
68. It is required to find the sum of the mth powers
of the roots of the equation a - by + cy2 = 0.
69. From the equation 1 -y + y2 =0, find the values of
the functions y", y-", ey, log y, sin y, sin-l y and tan )
70. Given the equation xy" - y + a =0, to find the values
of y2, y-2, y-, eY, log y, sin y, cos- y and tan-' 3y.
71. Find all the roots of the equation y3 - 9y +28 = o,
by means of the Theorem of Lagrange.
72. Given the equation 1 - y + ay= 0, to find the values
of yl, y~-", log y, siny and sin-' 2y.
73. Apply Lagrange's Theorem to deduce sin y from the
equation y= 1 - x sin y.
74. Find the values of u, sin u and sin mu in terms
of 0, from the equation 0 = u + e sin u.
75. Required the value of y from the equation
I + x sin y = tan y,
by means of Lagrange's Theorem.
76. Apply Lagrange's Theorem to find the value of z
in terms of y, from the equation ay3 4 zy' — a 23 = 0.
a(1 - e2)
77. From the equation r — a ---- e' it is required to
1 + e cos 0'
find the values of e and log r in terms of the rest.
78. Apply Laplace's Theorem to find the value of ey
from the equation y =log (z? +x sin y).
79. If u2 = (x - a)2 + (y - b)2 + (z - c)2, it is required
to prove that
( ) ( ) d2 (I)
+ + - ---- =o., dx' dy2 dz'