Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/159



Using the notation of differentials, formulas (25) and (26) may be written

Substituting the value of ds from (27) in (26),

An easy way to remember the relations (24)-(26) differentials dx, dy, ds is to note that they are correctly represented by a right triangle whose hypotenuse is ds, whose sides are dx and dy, and whose angle at the base is $$\tau$$. Then

and, dividing by dx or dy, gives (24) or (25) respectively. Also, from the figure,

the same relations given by (26).

In the derivation which follows we shall employ the same figure and the same notation used in §67

From the right triangle PRQ

Dividing throughout by $$(\Delta \theta)^2$$, we get