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

TRACING OF POLAR CURVES.

155

dr
do has, in order to see whether r is increasing or diminishing
dr~
with 0. When r is negative, -d =+, indicates a diminution in
the absolute numerical value of r, and vice versd.
It is sometimes necessary, in order to make out properly
the nature of the curve, to put down a few of the values of r
'ir $7r Wr $W
corresponding to such values of 0, as 2 7r r. - -,... &c.
When 0 = a makes r = 0, it is clear that there is a tangent at the origin making an angle a with the prime radius.
When 0 = a makes r - oo, we must find the corresponding
asymptote.
It is usual to neglect the negative values of r in tracing
polar curves, but there is no reason whatever to justify
this omission, and we shall therefore always consider these
values to be, as they really are, just as important as the
positive values.
224. In measuring positive and negative values along r, Positive
it will save mistakes, to suppose an arrow-head fixed upon the tive values
radius vector to revolve with it as 0 increases; then this arrow- ofr, how
measured.
head, supposing it to point in the positive direction when 0=0,
will always continue to point in the positive direction, whatever
be the value of 0. Thus, in fig. 24, let 0 = 7 and then from
S to P is the positive direction, and from S to P' the negative:
in fig. 25, let 0 = - +4, and then from S to P is the positive
direction: in fig. 26, let 0 = +,and then from S to P is
the positive direction: and in fig. 27, let 0 = - and then
4
from S to P is the positive direction.
Thus we see that the positive direction with reference to a
revolving line is not a fixed direction in space, but depends on