86 Mr Sang on the Proper Form 



it does not follow that whenever we find two axes which give 

 the same time of oscillation, the distance between those axes 

 must, of necessity, be the mean distance of oscillation ; for on 

 each side of the centre of gravity there may be found two 

 axes of motion, giving exactly the same time of oscillation ; 

 those axes being symmetrically placed. That position of the 

 knife-edge necessary to give a certain oscillation, is found by 

 resolving a quadratic equation, and in all possible cases this 

 quadratic has two roots, thus exhibiting the two positions of 

 the knife-edge ; and again, by supposing gravity to act in the 

 opposite direction, that is, on inverting the pendulum, there 

 appear other two roots. 



Now each solid has a maximum rapidity of oscillation, be- 

 yond which it is impossible to pass : when this, limit is just 

 about attained, the two roots of the quadratic equation ap- 

 proach each other and coalesce just at the limit, so that a con- 

 siderable change in the position of the fulcrum thereabouts 

 would produce a very small alteration on the rate of oscilla- 

 tion. It is quite clear, then, that experiments close to the 

 limit would be quite valueless in determining the length of 

 the simple pendulum. But quadratic functions have only one 

 state of maximum or of minimum ; and, in this case, since 

 there is a minimum degree of precision, there cannot be an 

 absolute maximum one. The limits offered by the mecha- 

 nical properties of matter are the only ones which give rise to 

 a best form. The general principle is this, that the farther 

 we can keep from the position of minimum exactitude, the 

 greater will be the precision attained to ; or, in other words, 

 we must seek for as great a disparity as possible between the 

 distances of the conjugate fulcrums from the centre of gravity 

 of the system. 



To see clearly the nature of the question which now opens 

 to us, we may put it in this shape. 



The mass of a pendulum being given, and the time of its 

 oscillation, it is required so to dispose of it as that the deter- 

 mination of the length, by means of two conjugate knife-edges, 

 may be the best possible. 



For this purpose, it is quite clear that the quantity repre- 



