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PROFESSOR J. CASEY ON A NEW 
or 
dv 
a$~ a ‘ 
Hence the envelope of LQ is a cycloid, and it is evident that Q is the point of contact. 
This is the cycloid that would be described by a fixed point in the circumference of the 
circle, whose diameter is the line IL, rolling on the line Z. 
Cor. From B let fall the perpendicular BW on the diameter VP of the rolling circle, 
then we have IW = IQ ; and therefore the locus of the point W is the envelope of VP, 
and it is the evolute of the cycloid described by Q. 
75. The circle whose centre is P, and which touches the line LX, also touches the 
locus of Q. Hence we have the following theorem : — 
If a variable circle has its centre on a given cycloid , and if it touches the tangent at 
its vertex , its envelope is another cycloid. 
76. If a variable circle touch a given cycloid , and also touch the tangent at the 
vertex , the locus of its centre is a cycloid. 
Fig. 10. 
Or we may give a direct proof of this last theorem : let the angle XLQ=^XLP = ^<p 
be denoted by 0 ; now we have 
i/=2a<p=4:aQ ; 
hence the envelope of LQ is a cycloid. Again, LP=LQcos0, but LP = 2asin20, 
.*. LQ=4« sin Q ; and therefore Q is the point of contact of LQ with its envelope, and 
the proposition is proved. 
77. If LP, L'P', L"P" be three fixed tangents to a variable cjcloid, we have 
v=2 a<f>, v'=2 a<p', v"=2 a$". 
Hence v'—v cp'— <p 
— , =-» — -7= constant. 
v'—v (p"—<p' 
Hence the tangent at the vertex of the cycloid is divided in a given anharmonic ratio by 
the three given tangents and the line at infinity. Hence we have the following theorem : — 
Being given three fixed tangents to a variable cycloid, the envelope of the tangent at 
the vertex is a parabola. 
78. If four fixed tangents to a cycloid be given, the tangent at the vertex is a common 
tangent to two parabolas. Now being given two parabolas they have, in addition to the 
