[♦339 ] 



The above folution aflEbrds a confpicuous Inftaiice of the 

 advantage of this method, in the ready manner in which the ge- 

 neral law of the feries is derived. This feries has been in- 

 veftigated by feveral authors fince Sir I. Newton, who firft 

 invented it. But all have only deduced a few of the firfl^ 

 terms, without any proof whatever of the law of the feries. 

 Indeed to have deduced by any of their methods even the lo^' 

 term would have been an almoft infuperable labour. 



Example III. To exprefs the hyperbolic logarithmic fecanc 

 by a feries afcending by powers of the arch. 



SojLUTiOxN. Let a, j-, and / be the arc, fecant, and logarithmic 

 iecant, rad. being unity. By the nature of the circle 



a ~ — ===L, and alfo / = — 

 Ws~ — I •*■ 



•.• /' — a'' * s' — ri, or taking the fluxions and making a 



conftant, 



2 / /= a' X 2SS = a- X is- >•• i or 7 = a' s' = /' + a'^ (A). 



But when a = o, s = t, •.• I — o and I— a ^/' — i = o: whence 



from the equation A it follows that all the uneven fluxions of 



/ are — o. becaufe any odd fluxion of the equation mufl: contain 



in each term the inferior odd fluxions of /. For the convfeniency 



U u 2 of 



