The resistance of the air, of course, has a great influence on a pendulum, and is one of the chief causes that bring it ultimately to rest. Even the variations of pressure of the atmosphere which the barometer shows as the weather varies have an effect on the going of a clock. Attempts have been made by fixing barometers on to pendulums with an ingenious system of counter balancing to counteract this, but these refinements are not in common use, and are too complicated to be susceptible of effective regulation. APPENDIX TO CHAPTER IV. It may be useful to give a simple form of proof of the law which governs the time of oscillation of a pendulum whose length is given. Unfortunately, it is impossible to give one so simple as to be comprehended by those who know nothing whatever of mathematics. It is, however, possible to give a proof that requires very little mathematical knowledge. We know that when a mass of matter is whirled round at the end of a string it tends to fly outwards and puts a strain on the string. The faster the speed at which the mass is whirled, the stronger will be the strain on the string. Suppose that the length of the string equals R, the velocity of the mass as it flies round equals V. Let _a_ be the body whirled round by a string _o a_ from a centre at _O_. The body always, of course, tends to fly on in a straight line from the point at which it is at any instant. But that tendency is frustrated by the pull of the string which constrains it to take a circular path. It is, of course, all one whether the force that tends to pull the body inwards towards _O_ is a string or an attractive force of any kind acting through a distance without any string at all. Evidently if the body keeps its place in the circle it must be because the centrifugal force tending to whirl it out is equal to the centripetal or attractive force tending to pull it in. [Illustration: FIG. 60.] The strain on the body, due to the force tending to pull it inwards, we shall designate by F, meaning by F the number of feet of velocity that would in one second be imparted to the body by the attractive force. Suppose that at some given instant of time the body is at a point _a_. At that instant its _direction_ will be along _a b_, tangential to the circle at _a_, and that is the path it would take if the centripetal or attractive force ceased to act just as the body got to _a_. In that case the body would be whirled off like a stone from a sling along the line _a b_, and would at the end of a given time, let us suppose a second, arrive at _b_. But it is not so whirled off; it is attracted towards _O_ and pulled inwards, and comes to _c_. Hence, then, the attractive force acting during one second must have been sufficient to pull the mass in from _b_ to _c_. But we know that if an accelerating force (F) acts on a body for a second it produces a final velocity equal to F at the end of the second, and an average velocity half F during the second.