672 ' PHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3. 



being supposed equal to unit, and its fluxion also equal to unit, let half the 

 sum of the extreme ordinates be represented by a, the difference of the first 

 fluxions of these ordinates by b, the difference of their 3d, 5th, 7th and higher 

 alternate fluxions by c, d, e, &c. then the area shall be equal to 



'^ - ll + 7lo - 35iio + lisieoo - ^""^ ^^'^h '' ^h« ^''^ th^°'*«'" ^°'" fi"d'"& 

 the area. The rest remaining, let a now represent the middle ordinate, and 



the area shall be equal a + A _ _|i_ + ^^ _ _^_ + &c. And this 

 is the theorem which the author makes most use of. When the several inter- 

 mediate ordinates represent the terms of a progression, the area is computed 

 from their sum, or conversely their sum is derived from the area, by theorems 

 that easily flow from these. 



These general theorems are afterwards applied for finding the sums of the 

 powers of any terms in arithmetical progression, whether the exponents of the 

 powers be positive or negative, and for finding the sums of their logarithms, 

 and thereby determining the ratio of the uncia of the middle term of a bino- 

 mial of a very high power to the sum of all the uncia;. This last problem was 

 celebrated among mathematicians some years ago, and by endeavouring to re- 

 solve it by the method of fluxions, the author found those theorems, which 

 give the same conclusions that are derived from other methods. They are 

 likewise applied for computing areas nearly, from a few equidistant ordinates, 

 and for interpolating the intermediate terms of a series, when the nature of 

 the figure can be determined, whose ordinates are as the differences of the 

 terms. 



In the last chapter, the general rules, derived from the method of fluxions 

 for the resolution of problems, are described and illustrated by examples. 

 After the common theorems concerning tangents, the rules for determining 

 the greatest and least ordinates, with the points of contrary flexure, and the 

 precautions that are necessary to render them accurate and general, which were 

 described above, are again demonstrated. Next follow the algebraic rules for 

 finding the centre of curvature, and determining the caustics by reflexion and 

 refraction, and the centripetal forces. The construction of the trajectory is 

 given, which is described by a force that is inversely as the 5th power of the 

 distance from the centre, because this construction requires hyperbolic and 

 elliptic arcs, and because a remarkable circumstance takes place in this case, 

 and indeed in an infinity of other cases, which could not obtain in those that 

 have been already constructed by others, viz. that a body may continually de- 

 scend in a spiral line towards the centre, and yet never approach so near to it 

 as to descend to a circle of a certain radius; and a body may recede for ever 



I 



