DR WALLACE ON A FUNCTIONAL EQUATION. 649 



and passing from the logai'ithms to the numbers, we obtain 



tan (45° + i ^) = tan (45° + i (p,) tan (45° + i cp^) tan (45° + ^ (^3) ... tan (45° + A (p„) . (9) 



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



tan(45°-i^)=tan(45°--i^,) tan(45°-i^2) tan (45°-^ ^3) . .. tan(45°-|0„). (10) 

 These formulae 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. 



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



and taking the square roots of the results, we find 



i _? 2V(/-«p) 



e« -e -= ^ '-; 



a 



(11) 



(12) 



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



We may also express xhj (p; for since 



i/ = a8ec(p, V (^^ — a^) = a tan ; 



and 1/ + ^ (y^ — a^) = a (sec(p + taa(p)=a tan (45° + i 0) ; 



^, „ « ivr I rtan(45° + i0)] f tan (45°+ 4 0)) 

 therefore — = Nep. log i ^ — | = m com. log j ^ — ^ \ ■ (IS) 



In these formulae, m=.43429448, and log »«= 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 (p for -^ , and making 



the parameter = 1, we have 



y d x = dta,n(p = sec' (p d cp ; 

 Now y=sec (p , 



therefore dx =sec <p . d (p= — J— ; 



cos (p 



and integrating, so that x and (p may begin together, 



2^== Nep. log ^ + "°^ 

 ^ ^ l-sm0 



VOL. XIV. PART II. 6 F 



