Page 146 - Foundations Of Differential Calculus

7. On the Differentiation of Functions of Two or More Variables 129
and when x replaces dx and y replaces dy,wehave

yx − xy
y 2 =0.

x
. Then
II. Let V = 2 2
x − y
2
−y dx − yx dy
dV = 3/2 ,
2
2
(x − y )
so that
2
2
−y x + y x =0.
2
2
(x − y ) 3/2
2

y + x + y 2
III. Let V = , which is a function of zero dimension in x
2
−y + x + y 2
and y. Then
2
2x dy − 2xy dx
,

dV =
2 2 2 2 2
x + y − y x + y
and when x and y are substituted for dx and dy, the result is zero.

x + y
IV. Let V =ln . Then
x − y
2xdy − 2ydx
dV = ,
2
x − y 2
and
2xy − 2yx =0.
2
x − y 2
√
x − y
V. Let V = arcsin √ . Then
x + y
ydx − xdy
dV = √ ,
(x + y) 2y (x − y)
and this formula enjoys the same property.

222. Now let us consider some other homogeneous functions and let V be
an n-dimensional function of x and y. Hence if we let y = tx, then V takes
n
the form Tx , where T is a function of t.Welet dT =Θ dt so that
n
dV = x Θ dt + nTx n−1 dx