650 PHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3. 



varies, and to have its direction changed into the opposite direction ; then ima- 

 gine a body to set out with a just velocity from a given point in the chain, and 

 to describe the curve. The tension of the chain at any point will be always 

 as the square of the velocity acquired at that point, and if a body be projected 

 with this velocity in the direction of the tangent, the curvature of the trajectory 

 described by it will be -f of the curvature of the chain at that point. We must 

 refer to the book for a fuller account of these and of other theorems. 



In the 13th chapter, the problems concerning the lines of swiftest descent, 

 the figures which among all those that have equal perimeters produce maxima 

 or minima, and the solid of least resistance, are resolved without computations, 

 from the first fluxions only. There are also easy synthetic demonstrations sub- 

 joined, because this theory is commonly esteemed of an abstruse nature, and 

 mistakes have been more frequently committed in the prosecution of it, than 

 of any other relating to fluxions. To give some idea of the author's method, 

 suppose the gravity to act in parallel lines, a to denote the velocity acquired at 

 the lowest point of the curve, and u the velocity acquired at any other point of 

 the curve. Suppose the element of the curve to be described by this velocity u, 

 but the element of the base to be always described by the constant velocity a. 

 Then it is easily demonstrated, without any computation, that the element of 

 the ordinate being given, the difference of the times in which the elements of 

 the curve and base are thus described is a minimum, when the ratio of those 

 elements is that of a to m ; i. e. when the sine of the angle, in which the ordi- 

 nate intersects the curve, is to the radius in this ratio. Supposing therefore 

 this property to take place over all the curve, the excess of the time in which it 

 is described by the body descending along it, above the time in which the base 

 is described uniformly with the velocity a, must be a minimum ; and this latter 

 time being given, it follows that the time of descent in this curve is a minimum. 

 When the gravity tends to a given centre, substitute an arc of a circle described 

 from that centre through the lowest point of the curve in the place of the base 

 in the former case ; and the property of the line of swiftest descent will be dis- 

 covered in the same manner. The nature of the line that among all those of 

 the same perimeter is described in the least time, is discovered with great faci- 

 lity, by determining from the former case the property of the figure when the 

 sum or difi^erence of the time in which it is described by the descending body, 

 and of the time in which it would be described by any given uniform motion, is 

 a minimum ; for the latter time being the same in all curves of the same length, 

 it follows that the figure, which has this property, must be described in less 

 time than any other of an equal perimeter. The general isoperi metrical pro- 

 blems are resoked, and the solutions are rendered more general, with like 



