[ 4«9 3 
the Theorems concerning the Refolution of a Fraction, 
that has a multinomial Denominator, into Fractions 
that have the Ample or quadratic Divifors of the Mul- 
tinomial for their feveral Divifors. Thcfe Dernon- 
ftrations are derived from the Method of Fluxions 
itfelf, without any foreign Aid ; or invariable Coeffi-] 
cients are determined by fuppoflng the variable Quan-’ 
tity or its Fluxions to vanifh. 
When a Fluent cannot be afligned by the Arcs of 
Conic Sections, it may however be meafured by their 
Arcs in fome Cafes ; and this may be confldered as 
the Third Degree of Refolution, or the Fluents may 
be called of the Third Order. On this Occafion fome 
Fluents are found to depend on the Rectification of 
the Hyperbola and Ellipfis, which have been formerly 
efteemed of an higher kind. The ConftruCtion of 
the elaftic Curve, with its Rectification, and the Mea- 
fure of the Time of Defcent in an Arch of a Circle, 
are derived from Hyperbolic and Elliptic Arcs; and 
the Fluents of this kind are compared with thofe of 
the Firft or Second Order by infinite Series. Becaufe 
there are Fluents of higher kinds than thefe, the Tra- 
jectories above-mentioned, which are defcribed by a 
centripetal Force, that is, as fome Power of the Di- 
ftance from a given Centre, when the Velocity of 
the Projection is that which would be acquired by 
an infinite Defcent, or by fuch a centrifugal Force, and 
the Velocity is fuch as would be acquired by flying 
from the Centre, are employed for reprefenting them. 
A Ample ConftruCtion of thefe Trajectories had been 
given above, by drawing Rays from the Centre to a 
Right Line given in Pofition, increafing or diminifh- 
ing the Logarithms of thofe Rays always in a given 
G g g 2 Ratio a 
