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(/'(<p) sin 2 <p) 
Vry r/ fflSlIKp 
* ’ /' (<p) sin 2 <p cos<p 5 
/'(<>) sin,< P=(^*> 
■ * ftp) f ‘ : J'gin 2 ^ (cos 
•••••. ( 71 ) 
If a be any even integer the integration on the right-hand side can be performed. 
See Williamson’s 4 Integral Calculus.’ 
(4) To find a curve whose tangential equation is the same as its intrinsic equation. 
Here we have/^p) sin <p4-§f'((p) cos <p d<p=f(<p), 
or % C fO) sin2 <t>) =f(<P ) sin <? ; 
/'($) sin 2 <p=C tan |i p, 
/(9)=«{tan 2 -|?)+logtan 2 ^)[, (72) 
where a stands for — • 
4 
(5) If the tangential equation of a curve be and the intrinsic equation 
s=f'(<p), find the curve. 
We ha ve/'(<p) sin <p +jf'(<P) cos <p d<p=f\q ) ; 
f"(<p) 2 cos <p 
/'(?>) _ 1 — sin (p* 
Hence f(<p)= 
Ci 
(1— sin <p) 2 ’ 
.-./(<p)=C 2 +C 1 
(73) 
(6) To find the intrinsic equation of the curve 
j4=l+(cot <p)K 
This is the curve whose ordinary tangential equation is 
or the curve whose trilinear equation is 
a "+/3 *-{-y * = 0. 
We havey'(<p) sin 2 <p= — (tam<p+l) 2 ; 
^ (/'(<P) sin 2 <p )= — § (cot* <p + cot 1 <p) sec 2 <p ; 
