FORM OF TANGENTIAL EQUATION. 
425 
But 
but 
x—(^—a) sin + & sin^-^)<p, 
3 /=(£— a) cos $-\-b cos^ ? -^^<p ; 
ds 9~ a / , n a / ia\ i 2 v'ab 
■••^=V( a +*) A ^ fl )> C = TTI' 
</0 g — 2 «’ 
•■■S=^ 2 (“+W9). 
••• s =/££+ 2 ( a + J ) E ( c ' 6 ); • 
s'= 2(« + ^)E(6-', fl) (see art. 83), 
(245) 
s' (246) 
The same result as before. 
CHAPTER VII. 
Section I . — Parallel Curves. 
121. The intercept which a parallel at the distance k from the movable line 
x-\-y cot <p — y = 0 at either side makes on the directing line is v±k cosec <p, the choice of 
sign depending on the position of the parallel with respect to the origin. Hence we 
have the following theorem : — 
If v—f(<p) be the tangential equation of a curve, the tangential equation of a parallel 
curve at the distance k is 
v=f(<p)± : k cosec <p (247) 
Thus the parallel to the parabola is 
v=a tan ty^rk cosec <p, (248) 
and the parallel to the cissoid 
(2a— *)®=27a 2 vcot 8 <p 
is 
) (2a— v) sin <p+k( 3 =27a 2 (v sin <p+#)cos 2 <p (249) 
122. By the method of art. 26 we get the coordinates of a point on the parallel 
curve to be 
A , =/'(<p)+/ , (<p)sin<pcos<p±^sin<p, (250) 
y— — /'($) sin 2 <p±# cos <p ; (251) 
