In the case shown in Fig. 73, however, the end cannot be reached in this way. For if the fourth point d (Fig. 79) pass to one side of P, and has its R. H. field to the left, and its L. H. field to the right of it, the portions Q b and P a of the line are covered by unlike fields, so that no turning can take place about their points, but the piece QP is covered with a pair of like fields, and remains a line of centres for left-handed turning. To make QP disappear, the normal to d must pass through P itself, in which case turn- ing can still take place about that point alone, which is the point of intersection of all the nor- mals, and is therefore common to the boundary lines of all the fields. In order to make turning impos- sible in such a case, a fifth point
must be added to those shown in Fig. 79, so as to cover dis- similarly (as shown in Fig. 80) the still remaining field of turning P Q. If the fourth point of restraint be so placed that P and Q coincide, two more points are necessary to prevent turning, one for left- and the other for right-handed turning.
Fig. 79.
The same thing holds good for the case shown in Fig. 71, where the normals to three points of restraint intersect at one point; for here also one additional point of restraint is required to pre- vent JR. H. and another to prevent L. H. turning. This and the foregoing case may be stated generally in the proposition: If the normals to three points of restraint of any figure cut one
