Page:Calculus Made Easy.pdf/79

${\dot {y}}$. If $x$ was the variable, then its fluxion was called ${\dot {x}}$. The dot over the letter indicated that it had been differentiated. But this notation does not tell us what is the independent variable with respect to which the differentiation has been effected. When we see ${\dfrac {dy}{dt}}$ we know that y is to be differentiated with respect to t. If we see ${\dfrac {dy}{dx}}$ we know that y is to be differentiated with respect to $x$. But if we see merely y˙, we cannot tell without looking at the context whether this is to mean ${\dfrac {dy}{dx}}$ or ${\dfrac {dy}{dt}}$ or ${\dfrac {dy}{dz}}$, or what is the other variable. So, therefore, this fluxional notation is less informing than the differential notation, and has in consequence largely dropped out of use. But its simplicity gives it an advantage if only we will agree to use it for those cases exclusively where time is the independent variable. In that case ${\dot {y}}$ will mean ${\dfrac {dy}{dt}}$ and ${\dot {u}}$ will mean ${\dfrac {du}{dt}}$; and ${\ddot {x}}$ will mean ${\dfrac {d^{2}x}{dt^{2}}}$.

Adopting this fluxional notation we may write the mechanical equations considered in the paragraphs above, as follows:

distance	$x$
velocity	$v={\dot {x}}$,
acceleration	$a={\dot {v}}={\ddot {x}}$,
force	$f=m{\dot {v}}=m{\ddot {x}}$,
work	$w=x\times m{\ddot {x}}$.

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