Sj H are two fixed points, and a curve is described such
that, if P be a point in it, the rectangle contained by SP
and HP is constant ; shew that the straight lines drawn from 8
at right angles to SP and from H at right angles to HP meet
the tangent at P in points equidistant from P.
Let Tj T' (fig. 87) be the points in which the straight lines
so drawn meet the tangent at P, SP= r, HP= r.
Then cosSPT-J, cosHPT=^,

cosfiPr dr 1

HP =m S.
cqbHPT' ^1? >

ds
but, since rr is constant^

dr , dr _
ds ds '



l£-4.] RIDERS. 161



or



r r

ds da

PT=PT.
6. Trace the curve whose equation is



- « 1 - tan0.
r



We find that



dO _ 1-tanfl ^ rffl _ g

r rfr"l+tan 8 « and ^ rfr " 1 + tan*0 '

dQ
when = 0, r = a, r -j- = 1, and the curve cuts the prime

radius at an angle - ;

when > < - ,

r is positive and changes from to oo, and when = — 1

r -T- = -, givmg an asymptote;



<?> 4 < 2'



r is negative and changes from oo to 0, and when = - ,

r* j- = 0, or the curve passes through the origin in a direc-
tion perpendicular to the prime radius;

2



> - < 7T,

r is positive and changes from to a, when = w, r j=l,



and the curve again cuts the prime radius at an angle —.
