376 
PEOFESSOE J. CASEY ON A NEW 
Section II . — Transformation of Polar into Tangential Equations. 
24. The polar equation of a curve being given, to find its tangential equation. 
Let the polar equation be g=F(Q), then 
Fig. 3. 
tan 4= 
F(0) 
E'(fl)- 
• ( 1 ) 
and 
Also we have v sin <p=% sin 4> that is, we have 
v sin ^5=F(0) sin 4 . . . . (2) 
$4 -<p+yp=r (3) 
Then eliminating d and 4 between equations 
(1), (2), (3). The result will be the tangential 
equation. 
Ex. Let the polar equation be 
g m =a m sin md 
We find, by taking logarithmic differentials, 
tan \|/= tan mQ ; 
.". ■p=mQ , 
and v m sin m <p — % m sin™ 4 = a m sin w+ 1 4 
Hence the tangential equation is 
(36) 
v sm ?>=«< sm 
m{ir — <p 
ttl- 1 - I 
or, putting <p in place of %— <p, 
v sm <p 
=al sin - m 9 X 
I m+ lj 
(3?: 
25. The family of curves represented by equation (36) includes several important 
species. The following Table contains the principal, with their corresponding tan- 
gential equations. 
Value of m. 
Name of curve. 
2 
Lemniscate .... 
— 2 
Equilateral hyp. 
l 
— 2 
Parabola .... 
! 
Cardioide .... 
1 
Circle 
Tangential equation of curve. 
v sin cj)=a ^sin 
" v sin <j) = a(sin 2 <p)% 
.or v—a*J 2 cot0 
v=— a cosec 2 0 
The parabola has another form of tangential 
equation, namely, v=atan0 
The director line in this form of equation is 
the tangent at the vertex. In the other 
forms it is the axis. 
1 sin ( p=a sin 2 \ 0 
(38) 
(39) 
(40) 
(41) 
(42) 
(43) 
(44) 
(45) 
