VOL. LXXIX.] PHILOSOPHICAL TRANSACTIONS. 571 



then will AXL + PXL'=nXM' and BXK-fQXK' = w. "" X m' ex- 

 press the law of the serieses. 



Cor. Let B = O, then the series ax + b^^^ + cx^ + dot' = a X (^ + 



Il2iJL*a:3 -j_ ^^' + ^2' ^ a;* -{- &c.), and the series 1 + p^ + az^ + rt' + &c. 

 = 1 4- p^ J 1. — X A- V X — - — or + -^i — ■ — — 37* + p X - — — — - — X* 



^~ 1.2 ^^^1,2.3 ~1.2.3.4 ~ *" ^ I .2.3... 5 



, (P' + A')^ « 



'1.2.3.. 6 a:" -|- &c.; the co-efficient of the term xf will be (p'* -f- a^) ^ or p 



« — I 

 X (p* + A^) ^ , according as n is even or odd. 



If in the equations before given for x be substituted a -=. h instead of a -f- ^, 

 then in the other quantities for h substitute — h. 



3. If in case 2 the difference between the two quantities (l -f pa + Qlo^ -f- 

 &c.) X (1 + pZj -h a^' + &c.) and (Aa -f- Ba* -f- ca* -|- &c.) X (aZI -|- bZ^* -f- 

 c^^ 4- &c.) is assumed = l-|-pX(aH-^) + aX (a + ^) + &c., then in the 

 serieses before given for A, b, c, &c. write respectively V— 1a, ■/— 1b, y^ — ic, 

 &c., and there will result the corresponding serieses. 



The same principles may be applied to many other cases. 



4. Equations of these formulae may be useful, when the sums of the serieses 

 correspondent to a value (a) of x are given, and the sums of the series corres- 

 pondent to a value (a •\- h) oi x is required, h having a small ratio to a: for 



instance, let the given series be x — -—- -f- - — - — — -| — - -f- &c.; the 



equation found in the first case is a -^ b — - ^ ^ ■ + g q "aT ~" ^^' = 



-&c.) + (l^^ 



— &c.); but a — -— -1- 



2.3.4.5 ^' 2.3 • 2 .3.4.5 



„ ^ — &c. are the sine s and cosine c of an arc a 



3.4 



of a circle whose radius is 1 ; and consequently, if the sine s and consine c of 

 an arc a be given, the sine of an arc a-{'b=z s X (1 "~2+5~4~~ ^^0 + c f'Z' — 

 --— -{- - — T— r— r — &c.), which series, if b be very small in proportion to a, 



converges much faster than the common series for finding the sine from the arc: 

 it has been given from different principles in the Meditationes, and is also easily 

 deducible from the series for finding the sine and cosine from the arc by the ' 

 propositions usually given in plane trigonometry: the cosine of the same arc 



- + i = c X (1 - :j^+ ,-|^ - &c.) - . X ri - rL + r:|^3-&c.) 



5. Let a quantity p be a function of x, or the fluent of a function of ar X '^, 



4d 2 



