Similarly in any other example in which one of the magnitudes is unlimited. Suppose I fling a stick at random in a horizontal plane against a row of iron railings and inquire for the chance of its passing through without touching them. The problem bears some analogy to that of the chessmen, and so far as the motion of translation of the stick is concerned (if we begin with this) it presents no difficulty. But as regards the rotation it is otherwise. For any assigned linear velocity there is a certain angular velocity _below_ which the stick may pass through without contact, but _above_ which it cannot. And inasmuch as the former range is limited and the latter is unlimited, we encounter the same impossibility as before in endeavouring to conceive a uniform distribution. Of course we might evade this particular difficulty by beginning with an estimate of the angular velocity, when we should have to repeat what has just been said, mutatis mutandis, in reference to the linear velocity. 7. I am of course aware that there are a variety of problems current which seem to conflict with what has just been said, but they will all submit to explanation. For instance; What is the chance that three straight lines, taken or drawn at random, shall be of such lengths as will admit of their forming a triangle? There are two ways in which we may regard the problem. We may, for one thing, start with the assumption of three lines not greater than a certain length n, and then determine towards what limit the chance tends as n increases unceasingly. Or, we may maintain that the question is merely one of _relative proportion_ of the three lines. We may then start with any magnitude we please to represent one of the lines (for simplicity, say, the longest of them), and consider that all possible shapes of a triangle will be represented by varying the lengths of the other two. In either case we get a definite result without need to make an attempt to conceive any random selection from the infinity of possible length. So in what is called the "three-point problem":--Three points in space are selected at random; find the chance of their forming an acute-angled triangle. What is done is to start with a closed volume,--say a sphere, from its superior simplicity,--find the chance (on the assumption of uniform distribution within this volume); and then conceive the continual enlargement without limit of this sphere. So regarded the problem is perfectly consistent and intelligible, though I fail to see why it should be termed a random selection _in space_ rather than in a sphere. Of course if we started with a different volume, say a cube, we should get a different result; and it is therefore contended (e.g. by Mr Crofton in the _Educational Times_, as already referred to) that infinite space is more naturally and appropriately regarded as tended towards by the enlargement of a sphere than by that of a cube or any other figure.