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532 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 
1 
= is the variation of this angle when @, varies 
n 
— cos 
vector, and cos *" 
n 
by an angle equal to a. Let this variation = ?; then if 0,— 0,'=6 
Now, if ~ increases, @ will diminish; and if m become infinite, 
a = + at = 0 when « and 8 are finite. 
Therefore, when mw is infinite, @ vanishes; therefore, the curve cuts the radius 
vector at a constant angle; therefore the curve is the logarithmic spiral. 
Therefore, if any curve be rolled on itself, and the operation repeated an 
infinite number of times, the resulting curve is the logarithmic spiral. 
Hence we may find, analytically, the curve which, being rolled on itself, 
traces itself. 
For the curve which has this property, if rolled on itself, and the operation 
repeated an infinite number of times, will still trace itself. 
But, by this proposition, the resulting curve is the logarithmic spiral; there- 
fore the curve required is the logarithmic spiral. As an example of a curve 
rolling on itself, we will take the curve whose equation is 
6 n 
f(=tena (cos ‘) 
n 
H a (sin 4 (cos oy 
a d a " : ~) *) 
r) 2n 
22” a2 (cos *) 
ao 22 py _——_—- 
as ae?) 2 6 2n—2 
a/ 920 a’ (cos i) ana + 22” @ (sin 2) (cos 2) 
n n 
f) ntl 
2-4 (cos *) 
n st 6 nt+1 
2 me eae a (cos *) 
NA (cos *) + (sin “) ie 
n n 

i 



6 6 
Now 6, — 4 = — cos! 22 = '= cos! cop = 
‘o n n 
n 
5 Se 
‘ eon 
substituting this value of 6, in the expression for r, 
x A 
n+1 n+1 
C52 a{ cos a 
m+ 


