VOL. LXXVI.] PHILOSOPHICAL TRANSACTIONS. 65 



according to the reciprocals of the great quantities, and it is done. If the fluent 

 of the quantity resulting be required, find it from the common methods, if pos- 

 sible; but if not, reduce the terms not to be found into an infinite series, and 

 then find approximate values to each of the terms, &c. M. Euler, and others, 

 reduced the series koc'' + bx'^-^* -\- ca:'+*' -|- &c. into a series a' sin. ra. -\- -& sin. 

 (?• + *) a + &c. &c. where a denotes the arc of a circle, whose sine is ax, &c. 

 It may be easily reduced into infinite other serieses proceeding according to the 

 dimensions of quantities, which are functions of x ; but it is most commonly 

 preferable to reduce it into serieses proceeding according to the sines, cosines, 

 tangents, or secants of the arcs of circle, which sines, &c. can immediately be 

 procured from the common tables. 



It has been observed in the first part, that to find the root of an equation, an 

 approximate value much more near to one root of the equation than to any 

 other must be given. In this part it is further observed, that serieses deduced 

 from expanding given quantities, so as to proceed according to the dimensions of 

 the unknown or variable quantities, will not converge if the unknown quanti- 

 ties be greater than the least roots of the above-mentioned equations ; and that 

 they will not converge much, unless the unknown quantities have a small pro- 

 portion to the least roots: and if the given quantities be expanded into serieses 

 descending according to the dimensions of the unknown quantities, then the se- 

 rieses resulting will not converge if the greatest roots of the equations before- 

 mentioned be greater than the unknown quantities ; and unless the unknown 

 quantities have a great ratio to the greatest roots the serieses will converge 

 slowly : for example, the serieses 



-|- ^y^ -\- &c. will never converge if x, z, or y, be greater than 1 ; but will 

 always converge when less than + 1 or + 1 -/ — 1 the least or only roots of 

 the equations 1 -f a; = 0, 1 — 3/* = 0, and 1 -f z^ = O. The series y -\- ^y^ 

 -|- &c. will always converge wheny is situated between -f 1 and •— ], in which 

 case alone it is possible. The same is true also of a series arising from expanding the 

 r(ax'" ■\- bx"'~^ -f- cx'"-^ -{• &c.)*'".f into a series proceeding according to the 

 dimensions of x, if the equation ax^ ■\- bx"' -\- cx'""'^ -f- &c. = O have only 2 

 possible roots a and — a, which are less in the manner before-mentioned than 

 any impossible root contained in it. 



If in either of the above-mentioned serieses the unknown quantity x, z, or y, 

 has a great proportion to 1, the series will converge very slow ; for example, if 

 a; = 1, ten thousand numbers at least are to be calculated, to procure the sum 

 of the series true to 4 figures ; therefore, in these and most other serieses, it is 

 necessary first to find a near value, viz. when x either = z, when e is very small ; 



VOL. XVI. K 



