2IO 



A NEW and UNIVERSAL SOLUTION 



Since 2 fin v X cof c zz fin 2)/, and 2 fin x X cof tt z:: fin 2r, 

 the two equations become, 



« — p iz: J fin 211, 



fin {« — w) zz '- fin 2T ; 

 take the fluxions of thefe equations, making n, v, and t vari- 

 able ; then, having brought « to fland by itfelf on one fide, we 

 fhall find, 



71 ZZ v (l + S Cof 21-), 

 « = 77 (l + 



e cof 2^^ \ 



cof 2': 



^V 



coi {n — ■!,)) 



whence, by equating thefe two values of w, 



;(l+aCof2v)=:.(i +^^f.^,;— ^-^^ 



Now V — T is a maximum when v — 'tt zz o, that is, when 

 > r: ff : therefore, if we divide both fides of the preceding equa- 

 tion by the equal quantities v and cr ; and further, reje(5l what 

 is common to both fides, and divide the remainders by s, we 

 fhall have, in the cafe when v — ■r is a maximum, 



r cof 2t 

 cof 2V — —-r-. T, 



col \n — ir) 

 If we combine thi^ equation with the two equations that ex- 

 prefs, in general, the relations of /; to v, and of n to t, we fliall 

 have three equations fufficient to determine the three arches, 

 «, f and T, in the cafe when v — it is a maximum. But, as 

 one of the equations is tranfcendental, this could only be done 

 by the method of infinite feries, and would lead into very per- 

 plexed calculations. We may,- however, by an eafy formula, 

 determine a limit, which the quantity v — t, when greatefl of 

 all, cannot exceed : this will be fufficient for the purpofe we 

 have at prefent in view, 



^ .From 



