380 
PROFESSOR J. CASEY ON A NEW 
Hence if c=/(<p) be the tangential equation of a curve, its intrinsic equation is 
s=f'(p) sin <p-\-§f'(<p) cos p dp (63) 
The result in equation (62) may be written in a form which in practice we shall find 
more useful. Thus 
ds d/p 
(/'(<?) si n2 <P) 
sm<p 
(64) 
31. Equation (63) may be established geometrically as follows: — Let LP, L'P' be two 
consecutive positions of the movable line, P, P' 
their points of contact with the envelope, and T 
their point of intersection. Let L'Q be a per- 
pendicular on LP. Now PP' is an element of the 
curve, and denoting it by ds, we have 
ds = P'T + TP = P'L' — QP = P'L' — PL + LQ 
= 6Z(LP)+IJ7 cos p=d(PL)-\-dv cos p ; 
.\ s=PL-|-jdj' cos p 
=f'(<p) sin <P+J/'(P) cos p d<p. 
Which is the same result as before. 
Fig- 5- 
Cor. 
s=PL-}-JPQ (see fig. art. 26); . 
••• !=ro+£(H.) 
• (65) 
• ( 66 ) 
From this it will be seen that the triangle LPQ is an important one in this theory. 
Observation . — The geometrical method of proof shows that this theorem holds even 
when the director line OL (see def. art. 2) is any plane curve; and we shall further on 
have to make use of this generalization. 
32. Before giving examples of the process of this section we will give the following 
integral reduction. 
J zaz 
to the normal form of elliptic integrals. 
Let z=\/3 cot 2 ijQ— 1, and 
+ ^ sin 2 0= A(0). Then after some easy reduc- 
tions, we get 
P zdz 2 C db 0 4 /tjP (1 + cos 0)^5 
J vrr? = ( V3-l)^3j W 6 J W 
Now 
cosO dQ A0 
J sin 2 6 . A (6) sin $’ 
9)-co«. A9, 
and 
