210 



A NEW and UNIVERSAL SOLUTION 



Since 2 fin v X cof v — fin 2v, and 2 fin r X cof »• — fin &*•, 

 the two equations become, 



n — p zz j fin 2c, 



fin (n — t) rr ^ fin 2t ; 

 take the fluxions of thefe equations, making ti f v, and r vari- 

 able ; then, having brought n to (land by itfelf on one fide, we 

 mail find, 



n — v (1 -f- s cof 2k), 



• / , , t Cof 2tt \ 



^ ' cof \n — *y 

 whence, by equating thefe two values of //, 



; (1 + a cof 2.) = ; (1 + CO f^l y ^ ). 



Now v — r is a maximum when v — tzo, that is, when 

 v = <x : therefore, if we divide both fides of the preceding equa- 

 tion by the equal quantities v and it ; and further, reject 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 — — p-7 r. 



COi {/I — ?r) 



If we combine this equation with the two equations that ex- 

 prefs, in general, the relations of n to v, and of n to <r, we fhall 

 have three equations fufficient to determine the three arches, 

 ;;, v and t, in the cafe when v — <r 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 — sr, when greateft of 

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

 have at prefent in view. 



2 From 



