GEOMETRY. 
564 
[790 
and it will be noticed how the form of the last equation puts in evidence the two 
asymptotes - = + % of the hyperbola. Referred to the asymptotes (as a set of oblique 
CL 0 
axes) the equation of the hyperbola takes the form xy=c\ and in particular, if in 
this equation the axes are at right angles, then the equation represents the rectangular 
hyperbola referred to its asymptotes as axes. 
Tangent, Normal, Circle and Radius of Curvature, &c. 
20. There is great convenience in using the language and notation of the 
infinitesimal analysis; thus we consider on a curve a point with coordinates (x, y), 
and a consecutive point the coordinates of which are (x+dx, y + dy), or again a second 
consecutive point with coordinates (x + dx + \ d?x, y + dy + \ d?y), &c.; and in the final 
results the ratios of the infinitesimals must be replaced by differential coefficients in 
the proper manner; thus, if x, y are considered as given functions of a parameter 0, 
then dx, dy have in fact the values dd, ^ dd, and (only the ratio being really 
material) they may in the result be replaced by ^This includes the case where 
the equation of the curve is given in the form y = <£ (x); d is here = x, and the 
increments dx, dy are in the result to be replaced by 1, 
infinitesimals of the higher orders d' 2 x, &c. 
dy 
dx 
So also with the 
21. The tangent at the point (x, y) is the line through this point and the 
consecutive point (x + dx, y + dy); hence, taking £, y as current coordinates, the 
equation is 
£ ~ x = V - y 
dx dy ’ 
an equation which is satisfied on writing therein £, y = (x, y) or =(x + dx, y + dy). 
The equation may be written 
dy 
dx 
being now the differential coefficient of y in regard to x\ and this form is applicable 
whether y is given directly as a function of x, or in whatever way y is in effect 
given as a function of x: if as before x, y are given each of them as a function 
of d, then the value of C ^- is = ^ ~ , which is the result obtained from the original 
dx du du 
form on writing therein , for dx, dy respectively. 
So again, when the curve is given by an equation u = 0 between the coordinates 
(x, y), then ^ is obtained from the equation ~ ^ ^ 6re ^ ^ more 
dx