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

TANGENTS, NORMALS, &C.

125

174. The normal at the point P is the line PG (fig. 8) Coeq
which passes through P at right angles to the tangent PT: its tion to the
Normal.
equation is therefore
dx
Y' - Y - (( - ).
175. If we take s to represent the arc BP (fig. 6), s' the Proposiarc BQ, and c the cord PQ; then ds2=dx2
+ dy2, s being the arc
S-S - C of the
---- s- *. ---.curve.
OG - IV C OG - t )
C -S (1 + ( ) since c2= (' -,)2 + (y - y):.
8t — 8?
Now when x' approaches x the limiting value of - is
00 - x
ds s' -s - y
- that of is 1 by Lemma IX, and that of -
dx c I - I
dy
is dy, hence by Lemma VIII, we have
ds dy 2
dx V1 +, or ds2= d'+ dy2
- X - 5' - S Cor.
176. We have cos PRX - = -, and x
C - s C ==ds cos/f,
when x' approaches x the limiting value of PRX is f, that of ds sin.
x' - d s - s
is- and that of is 1; therefore
s —8 ds c
dx
cos = d- or d=x ds cos,j.
y ds
/'-Y_ y'-Y s'-s
Similarly sin PRX= - = Y.
C — s C
dy.'. sin = ds;.d. dy=dssinl/.
ds '