298 
Proceedings of the Royal Society 
I) and C have come np E, and the hand encompasses the whole 
circle whose radius is BE. 
Imagine now that the point of contact C has passed beyond E 
into the position c ; the tangent takes the position dc^ so that the 
course of the hand is from F along the arc FDMc? and hack along 
s 
cfcE to E ; the halfdength being still made up of the three parts 
Ftf, dc, cE j hut in counting the length dc and cE must he regarded 
as subtractive. 
In this way the imaginary hand is shorter than the circumference 
of the wheel, by as much as, in the symmetric position, the actual 
hand had been longer. 
Beyond the limits f = ± Jtt, the mechanical arrangement fails to 
illustrate the properties of the above expressions, and therefore, 
regarding them as purely analytical formulae, we may proceed to 
examine their characters. 
Placing the arc f, representative of the inclination, along the line 
of abscissae, and setting up the values of I and V as ordinates, we 
