An elementary treatise on cubic and quartic curves, by A. B. Basset.

THE CASSINIAN.

165

Let SP =r, HP = r'; draw SK, HK' perpendicular to SP,
HP meeting the tangent at P in K, K'. Let SPK =, HPK'= b'.
K
Then rr'= a2- c2; and since dr/ds = cos, -dr'/ds= cos 4', we
obtain r sec f = r' sec q'; whence PK = PK'.
249. If the normal at P meet SH in G, and C be the middle
point of SH, the angles SPG and CPH are equal; also
SP2: HP2:: SG: HG.
Draw PC meeting SH in C, such that the angle CPH = SPG;
then
HC sin CPH sin SPG cos
r' sin C sin C sin C'
whence H( sin C = r' cos 0.
Similarly SC sin C = r cos 0' = HC sin C, since r' cos < = r cos /';
whence SC = HC. Therefore C is the middle point of SH.
Again, if SY, HZ are perpendicular to PK,
SP2= PK. PY; P2 = PK'. PZ,
and PK = PK' whence
SP2: HP2:: P: PZ,:SG: HG.
250. Straight lines are drawn from S, H and C perpendicular
to SP, HP, CP respectively, meeting the normal at P in L, M and
N; prove that if p be the radius of curvature at P,
I 1 I 1
PL PM PN p
Let PSH =, PHS = ', PCS= X, PGS=,f; then
ds
PL = r cosec = d= '
PM = d PN d
1 I ds dX'
whence PL + Pds
P-L + -M= ds ds '