DR WALLACE ON A FUNCTIONAL EQUATION. gg5 



■^ =2.3025851 X 1.1837775=2.725748 ; 



^„=Ay^^ 18-3436.46 ^^^,,3^ 



We have now found CB=a, the moduhis of the equilibrated curve (which is 

 also the parameter of the catenary), to be 18.3436 feet, and AE= Hr=?/„= 140.6343 

 feet. 



Logarithmic calculation by the second formula ■: 



Logarithms. 



a' = 6 0.7781513 

 ^;=46 1.6627578 



±=lL=cos (<^„ = 82° 30' 19") 9.1153935 



45°+^(/)„=86°15'9r' 



«oin. log. <^^"(-^^°+3'^o) =1.1837772 0.0732701 

 rad 



Nep. log 10=2.3025851 0.3622157 



^=2.725748 0.4354858 

 a 



a; =50 feet 1.6989700 



■o=a;^^ £^=18.343585 1.2634842 



— =.3270898 9.5146671 

 a 



y^ =y^ JL = 140.6344 2.1480907 



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 be ?/ = a ./(a;) ; here/(^) denotes the tabular 

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

 equilibrated cm-ve 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 



