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

126

TANGENTS, NORMALS, &C.

These for- 177. If it be allowable to argue according to the memulm may
be very thod of Leibnitz, spoken of in (84), we may suppose Q
aimply ob- taken so near P that the arc QP may be regarded as a right
the method line coinciding with the tangent at P, and PO, OQ, PQ taken
respectively equal to dx, dy, ds; these differentials being regarded as infinitely small: then we have immediately
dx=ds cos +, dy = dssin;
dy
and therefore dx2 + dy2 = ds2, and = tan I.
dot
Which formulae are all correct results in consequence of a compensation of errors such as that shewn to exist in (85).
Though this method of arriving at these formulae is not
sufficiently exact for the purposes of elementary demonstration,
it is useful in enabling us to remember and make out these
formrule.
Subtan- 178. The line MT (fig. 8) is called the subtangent,
gent, subnormal and MG the subnormal, PG the normal.
normal.
Hence
yd xt
subtangent = ycot\ - dy-'
dy
subnormal = ytan = yd
dx
yds
normal = ysec / = d(For examples of the use of these formulae see Appendix L.)
Proposi- 179. Since =tan 1-, we have
tion. dx
To determine d y
whether the a 1 d
concavity dx d d
of a curve 1d +
is turned d )'
upwards or