6^2 'JR WALLACK ON A FUNCTIONAL EQUATION. 



meter of the curve has been found, therefore the curve may be constructed by 

 co-ordinates either from the table, or by fhe geometrical construction given in 

 article 48. 



60. The problem may be othei-wise solved as follows : Putting x and/(.?;) to 

 denote the co-ordinates of a tabular catenarian arc, similar to the half of that 

 formed by the chain, and F (x) for the tabular arc, the parameter being unity, 

 let the angle made by a line touching the curve and the horizontal axis be ^ : 

 Put 2 C for the length of the chain in feet, and 2 D for the distance between its 

 points of support ; these are, by hypothesis, given numbers. 



By the nature of the catenary (art. 39), 



« = Nep. log { :^5^^!Jlti^ I , and F X =tan <^ : 



^. „ . ^ ivT 1 f tan (45° + i d)) 1 x D 



Therefore, cot <^ . Nep. log [—"^-^^^ } = ^-^^ -^ • 



Now N denoting any number, 



Nep. log N : Com. log N = Nep. log 10 : Com. log 10. 



Again, Nep. log 10 = 2.3025851 = ^^^^ , 



therefore, <^ must satisfy this condition ; 



Nat. cot <t, . Com. log ( ^Itl^^l } = 



tan(45° + i0)l_^^g^gD 



C 



The value of </> is to be foimd by successive trials in the trigonometrical tables. 



D 3 



In the example of this problem — ■ = — ; therefore, the angle (p must satisfy 



this condition 



Com, log tan (45° + ^^) ^ 2605767 

 Nat. tan (p 



which is nearly time when (f = 71° 55' 30" ; for 



log tan (45° + i ^) = tan 80° 5? 45" = .7984515 ; 



and Nat. tan (p = 3.064031 ; 



.7984515 

 and a^6403r = -^'^^^^- 



When (p is knovm, the things required may be found as by the other method. 



01. Problem II. — The span and height of an equilibrated arch are given: the 



roadway over it is to be a straight line : the parameter of the curve, which is 

 a line equal to its thickness at the cromi, is also given : to find the nutneral 

 values of ordinates to the curve. 



Let the figure bounded by the straight lines A'E, EF, FH', and the curve 



