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

INDEPENDANT VARIABLE.

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which are of course the same expressions as in the preceding
article.
If we suppose dy constant, we have d'y = 0, d3y = o,
and therefore
f2 - ) dyd v ) dy Idd3' dx - 3 (d'x)2}
dX3 d5..
and so on.
99. That variable whose differential is supposed to be Independconstant is called the independant variable; thus when we put antlehat.
d2y
f2"() = d, iv is the independant variable; when we put
dy d'V
f2(^)= - dy, y is the independant variable; and when
we put f(x) = d2ydv- dy d' neither x nor y is the indXv
dependant variable.
100. It is important to remember that d represents What dy
dtV represents
f"(x), only on the supposition that x is the independant deoenhat
variable, and therefore, though we may always put d"y = we consider
to be the
f"(xi) dxv when v is supposed to be the independant variable, independant variwe may not do so if this be not supposed. d"y, therefore, able.
does not always represent the same quantity; what it represents depends upon what quantity is considered the independant variable.
Thus suppose y = "2, then dy = 2xvdx, and d2y= 2dvi2, if Example.
dox be constant. But suppose that cV = z2, and;. y= 4;
then dy = 43dz, d"y = 12Z2dz2, if dz be constant; now since
v = z2, dx = 2zdz, and.'. 12 dz'd = 3drv2, and.'. d2y = 3di2.
Hence, when x is supposed the independant variable, dy =
2 d, and when z or ix is supposed the independant variable,
d2y=3dxV2: from which it is clear that d2y does not mean the
same thing when x is the independant variable, and when iv is.
101. It is often necessary to change the independant
variable in an expression containing differentials; this we may