1908. No. I. ON THE GRAPHIC SOLUTION OF DYNAMICAL PROBLEMS. 9 
are calculated, and the curves drawn upon paper. m is assumed so large 
that the curves on the diagram are not too far distant from one another; 
in some places it will generally be necessary to assume larger values for 
n than at other places. For the sake of clearness, it is recommended that 
all curves U = P be dotted lines for p odd, and continuous lines for p 
n 
even, and that the odd numbers, 1, 3, 5, 7, etc., be written in the spaces 
between the continuous curves, from U =o onwards, as shown in figure 4. 
Fig. 4. 
If the point M of the path falls upon one of the dotted curves, the 
radius of curvature then equals the portion D of the normal between the 
two nearest continuous curves multiplied by the number of the interval. 
If M falls upon one of the continuous curves, we have to multiply D by 
the even number lying between the numbers of the adjacent intervals. 
The graphic construction of the path may now be accomplished in 
the manner given by Lord Kelvin; but in addition to compasses, it is an 
advantage to use a narrow tape divided into millimetres, by which the 
centre of curvature can be quickly marked. By means of pins, this tape 
is attached to, and detached from, the points in the path and their centres 
of curvature alternately, as the construction progresses. The product 
pD may be found by the aid of a sliding-rule. 
More accurate results are obtained by substituting transparent gelatine- 
paper, cross-ruled, for the tape. 
Before proceeding to a treatment of the cases in which the method 
must be modified, we will consider the connection between the radius of 
curvature @ of an assumed path through the point M (see fig. 5), and the 
radius of curvature pg of a path through the same point, tangent to the 
