DR WALLACE ON A FUNCTIONAL EQUATION. 649 



and passing from the logarithms to the numbers, we obtain 



tan (45° + i <^) = tan (45° + ^ 0,) tan (45° + \ (p,) tan (45° + i ^s) • • . tan (45° + A (p„) . (9) 

 And because being any angle, tan (45 + 6) tan (45°— 0)=1 ; therefore 



tan (45° - i </)) = tan (45° - 1 0,) tan (45° - ^ (p^) tan (45° - ^ 0,) ... tan (45° - 1 (p„) . (10) 



These formulge express elegant properties of the catenary, which are not less 

 general and remarkable than properties of a circle, which are contemplated with 

 high satisfaction by geometers. 



£ —- 2 v 

 42. Because e" + e «=-^^; by subtracting 4 from the squares of these equals, 



and taking the square roots of the results, we find 



e<^ -e "= ^^ ^; 



Therefore 



a 



and — = Nep. log ^ — ^^^ ^ = m com. log ^ — ^^-^ L , (H) 



and because 2'=y-a^ and j/=V(a' + z'); 



therefore — = Nep. log ^ ^ = m com. log ^^-^ ^ , (12) 



a a ° a ^ ■^ 



By these formulae, oc may be found from either y or z. 

 We may also express ochj (p; for since 



t/ = asec(p, »J {^^ — a^) = at3bn<p i 



and y + ^ {./ — «') = ^ (sec + tan <p)=a tan (45° + ^ 0) ; 



^^ ^ « M 1 rtan(45° + i0)l f tan (45° + ^ 0) ) 

 therefore — = Nep. log \ ^ — l1 l = m com. log \ ^ — ^^ I . (IS) 



In these formulae, »^=. 43429448, and log m= 9. 6377843. 



43. The properties of the catenary which have been hitherto found are all ex- 

 pressed in finite terms ; some of them, however, may be expressed by series, 

 which have remarkable properties ; these we are now to investigate. 



Resuming the equation of article 12, and putting tan for -^ , and making 



the parameter =1, we have 



^ rf a; = (/ tan = see' fi? ; 

 Now i/=seG<p, 



therefore d x=sec (p . deb = — J-- ; 



COS0 



and integrating, so that x and may begin together, 



2;. = Nep.log-l^t£I^; 

 ^ * l-sm0 





 VOL. XIV. PART II. 6 F 



