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

36

d2y x dy y
80. Change - + - =0, where x is the
de2 1-x dx 1 - v2
independent variable, so that 0=cos-lx shall be the independent
variable.
d4y 4 d'y 12 d2y 24 dy 24
81. If d - + --- — + y = 0, where
dxv4 z d x3 xZ~ dx x dx i4
x is the independent variable, find the equation wherein z=logi
is the independent variable.
d2 y dy
82. Reduce the equation a + b - + cy- 0, where
d ci dx_
dx is considered constant, into an equation wherein V/dx2 + dy2
is constant.
83. If a v/(d V)2 -+ (d y) = ds', where ds = V/dv + dy'
is invariable, it is required to find the corresponding formulae
when x and y are respectively the principal variables.
d2x d3x
84. If ydx be invariable, and for —, 39 &c., there
d '' di"v
be substituted - p2_-qy
be substituted - - y, y -, &c., a homogeneous function
of an order above the second, wherein neither x nor y is thS
independent variable, will become a function of y and its
differential coefficients relatively to x: required a proof.
85. Apply the formula d d dy to find the diffIrendx dy dx
cos x ( cos x)
tial coefficients of u = (ax) csx and a(Cs)
86. From the same formula, if y = a (2 cos 9 - cos 20) and
x = a (2 sin 0 - sin 20), deduce the values of dy and y.
87. If =f(td and ), it is required to express
87. If v =f (t) and y= (t, it is required to express
dy d2y
*' d2, &c. in terms of t.
dTX d"'
The last ten Examples are illustrative of the principles
explained in Articles (84) and (85).