56 KANSAS UNIVERSITY QUARTERLY. 
the co-ordinates of the portion of the curve limited by the newly 
determined axes A’N and A’M. 
And if a pair of a certain ratio be desired, it can be had at once 
by drawing a line radial from A’, so that any ordinate of it, as 
RW, is to the corresponding abscissa, A’W, as the given ratio. 
Then the co-ordinates of the point T, where this line cuts the 
curve, are the desired pair of radii. 
SOLUTION COMPLETE. 
By way of summary, we may now review the five predetermined 
conditions to which the solution was to comply, and see that each 
is fully met. 
I. We may deal with any distance between centers and use any 
desired scale; for the rectangle, Fig. 4, can be constructed with 
AB equal any value of d, and the curve BFE drawn at once, with 
a circle arc, the center of which is at C, AC being the diagonal of 
a square, the sides of which are ;; AB. 
II. To secure one radius, and to find another, corresponding 
for a given ratio, simply draw a line AK from A, inclined so that 
any ordinate GJ is to its abscissa AJ as the desired ratio. Lay off 
AH equal to the given or assumed radius, anu HK is the other of 
the pair. 
III. Now to fix the length of the belt for this pair, and to 
determine the new origin, draw, as already fully explained, KS, a 
45°-line, and SA’ parallel to KA, when A’ is the new origin, and 1 
the half length of the belt for this pair. 
IV. All other pairs of radii which will run to this 1, are now 
shown by the co-ordinates of the points of the part of the curve. 
included between N and M. 
V. To choose a certain pair, having a definite desired ratio, 
simply draw from the new origin A’, A’R inclined so that any 
ordinate RW is to its abscissa A’W as the desired ratio, when 
A’L and LF are the radii sought. 
We see, then, that every condition to which the solution was to 
conform is completely and exactly met in a very simple manner, 
making a fairly ideal treatment of this, an unusually troublesome 
problem. 
