When a plane frame which is just rigid is subject to a given system of equilibrating extraneous forces (in its own plane) acting on the joints, the stresses in the bars are in general uniquely determinate.
In fact in a unicursal curve the co-ordinates of a point are given as proportional to rational and integral functions of a parameter, so that any point of the curve is determined uniquely by means of this parameter; that is, to each point of the curve corresponds one value of the parameter, and to each value of the parameter one point on the curve; and the (a, t3) correspondence between the two points is given by an equation of the form MO, I) u (4), 01 3 =0 between their parameters 0 and 4); at a united point 4)=0, and the value of 0 is given by an equation of the order a+ 0.
It follows that the single resultant to which the system in general reduces is uniquely determinate, i.e.
We have seen that the stresses produced by an equilibrating system of extraneous forces in a frame which is just rigid, according to the criterion of 6, are in general uniquely determinate; in particular, when there are no extraneous forces the bars are in general free from stress.
Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity.
The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF ...
Thus a fdrce can be uniquely resolved into two components acting in two assigned directions in the same plane with it by an inversion of the parallelogram construction of fig.
The volcanic d'Entrecasteaux Islands are mostly larger, more elevated (the highest being 8000 ft.), and stand in deeper water than the Louisiade group. To the east of Kiriwina (Trobriand) lies a small group of uniquely formed islets, each of which is completely surrounded by a steep forest-clad marginal rampart of coral 300 to 400 ft.