670 PHILOSOPHICAL TRANSACTIONS. [aNNO J 742-3. 



fluents to others of a more simple form, when they are not assignable by a 

 finite number of algebraic terms. When a fluent can be assigned by the qua- 

 drature of the conic sections, and consequently by circular arcs or logarithms, 

 this is considered as the 2d degree of resolution; and this subject is treated at 

 length. An illustration is premised of the analogy between elliptic and hyper- 

 bolic sectors formed by rays drawn from the centres of the figures; the pro- 

 perties of the latter are sometimes more easily discovered, because of their rela- 

 tion to logarithms, and lead us in a brief manner to the analogous properties 

 of elliptic sectors, and particularly to some general theorems concerning the 

 multiplication and division of circular sectors or arcs. When 2 points are as- 

 sumed in an hyperbola, and also in an ellipsis, so that the sectors terminated 

 by the semi-axis, and the 2 semi-diameters, belonging to those points, are in 

 the same given ratio in both figures, then the relation between the semi-axis 

 and the 2 ordinates drawn from those points to the other axis, is always defined 

 by the same, or by a similar equation in both figures. This proposition serves 

 for demonstrating Mr. Cotes's celebrated theorem, as it is extended by M. De 

 Moivre, by which a binomial or trinomial is resolved into its quadratic divisors, 

 and various fluents are reduced to circular arcs and logarithms. The demon- 

 strations are also rendered more easy of the theorems concerning the resolution 

 of a fraction, that has a multinomial denominator, into fractions that have the 

 simple or quadratic divisors of the multinomial for their several denominators. 

 These demonstrations are derived from the method of fluxions itself, without 

 any foreign aid; the invariable coefiicients being determined by supposing the 

 variable quantities or its fluxions to vanish. 



When a fluent cannot be assigned by the areas of conic sections, it may 

 however be measured by their arcs in some cases; and this may be considered 

 as the 3d degree of resolution, or the fluents may be called of the 3d order. 

 On this occasion some fluents are found to depend on the rectification of the 

 hyperbola and ellipsis, which have been formerly esteemed of a higher kind. 

 The construction of the elastic curve, with its rectification, and the measure 

 of the time of descent in an arch of a circle, are derived from hyperbolic and 

 elliptic arcs; and the fluents of this kind are compared with those of the first 

 or second order by infinite series. Because there are fluents of higher kinds 

 than these, the trajectories above-mentioned, which are described by a centri- 

 petal force, that is as some power of the distance from a given centre, when 

 the velocity of the projection is that which would be acquired by an infinite 

 descent, or by such a centrifugal force, and the velocity is such as would be 

 acquired by flying from the centre, are employed for representing them. A 

 simple construction of these trajectories had been given above, by drawing rays 



