666 



DR WALLACE ON A FUNCTIONAL EQUATION. 



y^ = af(a\^) to be each 140.0343 feet; these have the same amplitude as ?// = 4fi 

 feet, the ordinate of the equilibrated curve. 



The following Table shews the length of forty-five ordinates at as many 

 points of the arch on either side of the crown. The first ten stand at equal dis- 

 tances of 1.834 feet along the roadway ; the remainder are distant from each by 

 half that extent, viz. .917. 



Co-ordinates of an Equilibrated Arch. 



The first two columns of the table express the length of the co-ordinates of 

 a catenary whose parameter is unity ; these are just the numbers of our second 

 table. The third column contains the values of the numbers in the first column 

 reduced to feet, by multiplying each by the number ff = 18.343585. and putting 

 down the results true to thousandth parts of a foot. 



The second column, or values of y reduced to feet by multiplying each num- 

 ber by a, would express the ordinates of the catenary ; and any ordinate {a y) of 

 the catenary, having to the corresponding ordinate of the equilibrated arch the 

 ratio of a to a', that is, ay : if = a: a', it follows that a a' y - a if, and ?/' = a' y. 

 Now a' = 6, therefore the numbers in the fourth column are found from those in 

 the second by multiplying each by G. 



The numbers in this table have the general properties which belong to the 



