378 
PROFESSOR J. CASEY ON A NEW 
29. Professor Cayley considers a “ curve as described (see Salmon’s ‘ Higher Curves,’ 
second edition, p. 33) by a point which moves along a line at the same time that the 
line revolves round the point. There is, then, this peculiarity at a point of inflection, 
the line first becomes stationary and then reverses the sense of its motion.” From this 
it follows that the line x-\-y cot <J5 — v will cut off a maximum or minimum intercept on 
the director line w 7 hen it passes through a point of inflection, and also it will make in 
the same case with the same line a maximum or minimum angle. Hence when 
x-\-y cot<p — v 
is an inflectional tangent, 
maximum or minimum 
and 
<p= maximum or minimum. 
Examples. 
(1) If a line of constant length slide along two rectangular lines, to find its envelope. 
In this case we have evidently 
v=a cos <p ; ,\ f\(p)=a cos <p. 
Hence from equations (46), (47) vre get 
x~a cos 3 <p, y= asirdcp; 
x%-\-y^=a? 
(2) If from any point in an ellipse perpendiculars be let fall on the axes, find the 
a 2 
envelope of the line joining their feet. In this case y(<p) = -^=====, and the 
required equation is 
ipyij I j wi -f - 1 
(3) Let j/=£tan m <p; then if we put - — ™ — k=a , we get the Cartesian equation 
x m+1 =ay m (53) 
(4) If i>=cjl + (cot <p)™j- 2 5 the Cartesian equation is 
-L. , . -L- (54) 
X 2-m^_y2-m — C 2-m \ ' 
Compare equation (51). 
Cor. If in this example we substitute — 2 n for m, we get 
1 1 1 
tfn+l-^-y n +l — C n+l 
( 55 ) 
