MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 525 
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By substituting for r, in the first equation, its value, as derived from the 
second, we obtain 
= Gy LG8)' 1-09) 
3 d Ys dy, ad 6, 
oes 
2 
If we know the equation to the rolled curve, we may find ( aa ) in terms 
of r,, then by substituting for 7, its value in the second equation, we have an 
: tA d 5 “ 
equation containing w, and aa from which we find the value of a in terms 
3 3 
of w,, the integration of this gives the equation of the traced curve. 
As an example, we may find the curve traced by the pole of a hyperbolic 
spiral which rolls on a straight line. 
The equation of the rolled curve is 6, = = 
dx.\ 2 dxz.\? Lat dx.\ ” 2 
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“ (75) Ge) +1]=3(G2 +1] 
d2.\ ? dx.\? 
aS =«3[ 5) +1] 
dy, : d ¥; 
d x, = 45 
dy; es ne a =u ae 
This is the differential equation of the tractory of the straight line, which is 
the curve traced by the pole of the hyperbolic spiral. 
By eliminating w, in the two equations, we obtain 
dr, d x, 
ce) 
This equation serves to determine the rolled curve when the traced curve is 
_ given. 
As an example we shall find the curve, which being rolled on a straight line, 
traces a common catenary. 
Let the equation to the catenary be 

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