92 BROOMALL : 



By physical curves, as here mentioned, are meant all those 

 mathematical curves which are found to be closely related to 

 certain physical phenomena. It is indeed surprising to note 

 how many such phenomena may be studied by the aid of a 

 few mathematical conceptions. Let us consider some of the 

 commoner relations one by one. 



The Ellipse. — The principal occurrence of the ellipse in 

 physical phenomena is as the type of planetary motion. The 

 orbits of the earth and other planets, as well as the orbits of 

 the periodic comets, are all ellipses, the sun being at one of 

 the foci of the curves. Any body under the iniiuence of a 

 central force which varies inversely as the square of the dis- 

 tance from the centre of force will describe an ellipse. In 

 this case the centre of force is at one of the foci of the curve. 

 If the body moves under a central force which varies directly 

 as the first power of the distance, then likewise is an ellipse 

 described, but in this case the centre of force is at the inter- 

 section of the two axes of the curve. This is the law of mol- 

 ecular physics and is of great importance in optics, acoustics, 

 etc. 



The Cycloid. — The cycloid in its simplest form is the curve 

 described by a point on the circumference of a circle which 

 rolls along a given straight line. In its broader sense the 

 cycloid may be defined as the curve described by any point 

 on the radius of a circle, or on the radius produced, when 

 such circle rolls upon a straight line or upon or within a 

 circle. It is seen from this that the very mathematical con- 

 ception of the cycloid is physical in its nature. It has, how- 

 ever, certain very real physical relations, two of which are 

 indicated by the names brachistochrone and tautochrone, 

 sometimes applied to it. The first of these, as its derivation 

 indicates, is the curve of quickest descent. In other words, 

 if we have a body constrained to fall down a frictionless 

 curve shaped like an inverted cycloid, for example, a wheel 

 or marble rolling down a track of sonic kind, it will take less 

 time for the bodv to roll down this curve from one level to 



