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PROFESSOR J. CASEY ON A NEW 
that is, 
§/ f (0) sin 
*’■ s ~ s+m 
-|-J cos Qdor ; 
s 
_/'(#)/ i(?))sin9 
“ /'(0)+/'i(?) 
+J cos 6f'(Q)dQ 
• ( 216 ) 
If we denote the angle which LP makes with OR by \f/, we have evidently 
d^=d<p+d0, . . (217) 
and we have also 
m=m- .......... (2i8) 
Hence eliminating 6 and <p between the three last equations, there will be a resulting 
equation between s and 4/, say 
s=m, (219); 
and this will be the intrinsic equation of the envelope of LP. 
94. Let the directing curve be the catenary, and let the functional symbols f,-fi be 
the same; then, since tr=f(Q)=f 1 (<p), we have 0=<P; .*. -4/= 20. 
Now we have, from the intrinsic equation of the catenary, 
/'j(<P )=c sec 2 <p ; .*. f'(Q)=c sec 2 A 
Hence, making these substitutions in equation (216), and putting 0=<p=^, we have 
the required intrinsic equation 
s=c|^sec|tan|+log ^sec|+tan|^|. ..... (220) 
95. Let the directing curve be the cycloid, and let, as before, f,f be the same, then 
we get the intrinsic equation 
s=2a(l— cos \|/), (221) 
a curve which we shall find to be a parallel to the cycloid. 
96. The most interesting application of our general equation is where the directing 
curve is a circle, and the relation between a and 0 is linear ; that is, 
<r =&_,(#), 
where is a constant. It will be seen that in this simple case the envelope belongs 
to the class of curves known as epicycloids and hypocycloids, and it will belong to one 
or the other according as tLj is positive or negative, or, what comes to the same thing, 
according as 0 is positive or negative. 
97. Let and let the radius of the directing circle ALC be g, then, from art. 93, 
\=l+b - (222) 
