DR WALLACE ON A FUNCTIONAL EQUATION. ^^5 



-^ =2.3025851 X 1.1837775=2.725748 ; 



a 



^„ = A,. =i^^i^^ =140.6843. 



We have now found CB=«, the modulus of the equilibrated curve (which is 

 also the parameter of the catenary), to be 18.3436 feet, and AE= HF=?/„= 140. 6343 

 feet. 



Logarithmw calculation by the second formula i 



Logarithms. 



a' = 6 0.7781513 

 ^;=46 1.6627578 



— = — = cos (</), = 82° 30' 19") 9.1153935 

 45° + ^0, = 86°15'9r 



<om. log. ^^^"(45°+^0J ^1.1837772 0.0732701 

 rad 



Nep. log 10=2.3025851 0.3622157 



^=2.725748 0.4354858 

 a 



x=m feet 1.6989700 



«=«_._ ^ = 18.343585 1.2634842 

 a 



— =.3270898 9.5146671 

 a 



^ =y _1 = 140.6344 2.1480907 



a 



In the catenary, we have now its parameter a = 18.343585 feet ; and, to 

 construct it, we may set off from C both ways, in the line EF, distances each 

 equal to a, and divide each of these into 100 equal parts. If now x denote the 

 number of these divisions between C and any point in the scale CE, the ordinate 

 of the catenary at that point will h^y = a .f{x)\ here /(«') denotes the tabular 

 value of the ordinate whose amplitude is x. The corresponding ordinate of the 

 equilibrated curve will be a' .f{^), for then y:y' — a\a'. There is, however, no 

 necessity for actually constructing the catenary ; it is merely subsidiary, and it 

 has been introduced here only as a geometrical representation of the relation be- 

 tween the tabular co-ordinates ^ and/(«). We have found its extreme ordinates 



VOL. XIV. PART Hi 6 K 



