Page:A Source Book in Mathematics.djvu/659

 Multiplication: $d\overline{vx}=xdv+vdx$,

or by placing

$y = xv$,

$dy = xdv + vdx$.

. . . . . . . . ..

Yet it must be noticed that the converse is not always given by a differential equation, except with a certain caution, of which [I shall speak] elsewhere.

Next, division:

$d\frac{v}{y}=\frac{\pm \;vdy\mp \;ydv}{yy}$

(or $$z$$ being placed equal to $$\frac{v}{y}$$)

$dz=\frac{\pm \;vdy\mp \;ydv}{yy}$

Until this sign may be correctly written, whenever in the calculus its differential is simply substituted for the letter, the same sign is of course to be used, and $$+dx$$ [is] to be written for $$+x$$, and $$-dx$$ [is] to written for $$-x$$, as is apparent from the addition and subtraction done just above; but when an exact value is sought, or when the relation of the $$z$$ to $$x$$ is considered, then [it is necessary] to show whether the value of the $$dz$$ is a positive quantity, or less than nothing, or as I should say, negative; as will happen later, when the tangent $$ZE$$ is drawn from the point $$Z$$, not toward $$$$A, but in the opposite direction or below $$X$$, that is, when the ordinates $$z$$ decrease with the increasing abscissas $$x$$. And because the ordinates $$v$$ sometimes increase, sometimes decrease, $$dv$$ will be sometimes a positive, sometimes a negative quantity; and in the former case the tangent $$IVIB$$ is drawn toward $$A$$, in the latter $$2V2B$$ is drawn in the opposite direction. Yet neither happens in the intermediate [position] at $$M$$, at which moment the $$v$$’s neither increase nor decrease, but are at rest; and therefore $$dy$$ becomes equal to $$0$$, where nothing represents a quantity [which] may be either positive or negative, for $$+0$$ equals $$-0$$; and at that place the $$v$$, obviously the ordinate $$LM$$, is maxi- mum (or if the convexity turns toward the axis, minimum) and the tangent to the curve at $$M$$ is drawn neither above $$X$$, where it approaches the axis in the direction of $$A$$, nor below $$X$$ in the contrary direction, but is parallel to the axis. If $$dv$$ is infinite