[ 33^ ] 



Example II. Given the fine of an arch (A) to find the fine 

 of /i times that arch. 



Solution. Let x = fine o£ A, y = fine of « A. Then 

 ~ -T or jy^ ^ I — X- = n- X- x i — -j-, taking the 



fluxions generally making x flow uniformly, and dividing hyj. 



J X. 1 — .V- —J XX = — n' x^j'. 



The m — 2 fluxion of this equation being taken by Prob. i 

 and s, when x — andj = 0, 



J + ?fi — 2. w — 3 J X — 2 A= — m — 2j x^ = — Ti- x'-y 

 2 



orj'=: x'^ y >■ n^ — ni — 'i\ ' • 



Now becaufe when x and j = 0, ^ = w r and 7 = 0; it follows 

 therefore that all the even fluxions of>' are ^ ; and taking for 

 m the odd numbers 3, 5, 7, &c. and a- for x, we have 



y = y + -y— + &c. =«x a x' + -^ Z-.^ &c. tf, ^, 



■^1.2 2. 3 4-5 



&c. being the preceding terms. Alfo if k and / be the p — i^ 



;> ~ — 2 p <{! ^ 



and /> terms /= (becaufe /^2= ZP — 1) = — k x- ^ — — ~ ^~ . 

 •r \ -f ^ 2/> — 2 '^ 2/ — I 



The 



