194 PHILOSOPHICAL TRANSACTIONS. [aNNO I/IS. 



equal distances: and in an example at the end of this proposition I have 

 showed how easily Sir Isaac Newton's Series, for expressing the dignity of a 

 binomial, may be found by this incremental method. The 14th proposition, 

 shows in some measure how this method may be of use in summing up of 

 arithmetical serieses. In the 15th proposition I show by some examples how 

 the proportions of the fluxions are to be found in geometrical figures ; from 

 whence immediately flows the method of finding the radii of their inosculating 

 circles, the invention of the points of contrary flexure, and the solution of 

 other problems of the like nature. In the l6th Proposition I show how the 

 method of fluxions is to be applied to the quadrature of all sorts of curves. 

 In the following Proposition I give a general solution of the problem of the 

 Isoperimeter, which has been treated of by the two famous mathematical 

 brothers the Bernoullis. In the 18th Proposition I give the solution of the 

 problem about the catenaria, not only when the chain is of a given thickness 

 every where, but in general, when its thickness alters according to any given 

 law. In the following Proposition I show the fornix, or arch which supports 

 its own weight, to be the same with the catenaria. In the two next Proposi- 

 tions I show how to find the figures of pliable surfaces which are charged with 

 the weight of a fluid. In the 22d and 23d Propositions I treat of the motion 

 of a musical string, and give the solution of this problem : to find the number 

 of vibrations that a string will make in a certain time, having given its length, 

 its weight, and the weight that stretches it. This problem I take to be entirely 

 new, and in the solution of it, in the last part of Prop. 23, there is a remark- 

 able instance of the usefulness of the method of first and last ratios. The 

 24th Proposition gives the invention of the centre of oscillation of all bodies; 

 and in the 25th Proposition I have given the investigation of the centre of 

 percussion. It is known that this problem is solved by the same calculus as 

 the foregoing ; therefore it is generally thought that these two centres are the 

 same. But that is a mistake, because the centre of oscillation can be only one 

 point; but the centre of percussion be any where in a certain line, which this 

 Proposition shows how to find. There is an error in this Proposition, which 

 I was not sensible of till after the book was published, therefore I take this 

 opportunity of correcting it. It does not affect the reasoning by which I 

 find the distance of the centre of percussion from the axis of rotation ; but it 

 is this, that I supposed the centre of percussion to be in the plane passing 

 through the centre of gravity, and perpendicular to the axis of rotation: 

 which is a mistake. It is corrected by the following Proposition. 



