g(j2 ^^ WALLACE 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 the geometrical construction given in 

 article 48. 



60. The problem may be otherwise solved as follows : Putting a; and /{a;) to 

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

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

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

 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), 



X = Nep. log I ^° ^ ' ^^^ I , and F x =tan (f) : 



rw,. » . ^ AT , r tan (45° + ^(p)] x D 



Therefore, cot . Nep. log {— ^^^^ } = jt^^ -^ ■ 



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> ■ Cora, log | ^^" ^^^^° + ^ *^^ | = .43429448 ^ . 



The value of ^ is to be found by successive trials in the trigonometrical tables. 



JD _ 3 

 C ~ 5 



D 3 



In the example of this problem — = — ; therefore, the angle ^ must satisfy 



this condition 



Com.logtan(45o + i0) ^ .2605767 

 Nat. tan 



which is nearly true when ^ = 71° 55' W ; for 



log tan (45° + i 0) = tan 80° 57' 45" = .7984515 ; 

 and Nat. tan = 3.064031 ; 



and n^^^^\^ ^ 26059 . 



^^^ 3.064031 



When is known, the things required may be found as by the other method. 



61. Problem II. — The span and height of an equilibj^ated arch are given : the 

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

 a line equal to its thickness at the crown, is also given : to find the numeral 

 values of ordinates to the curve. 



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



