^Q 



DR WALLACE ON A FUNCTIONAL EQUATION. 



p^ = af{x^ to be each 140.6343 feet; these have the same amplitude as ?// = 46 

 feet, the ordinate of the equihbrated 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 .884 feet along the roadway ; the remainder are distant from each by 

 half that extent, viz. .917. 



Co-ordinates of an Equilibrated Arch. 



Tabular Co-ordinates 



Co-ordinates of Arch in 



Tabular 



Co-ordinates 



Co-ordinates of Arch in 



of Catenary. 



Feet. 



of Catenary. 



Feet. 



X 



y 



X 



y' 



X 



y 



X 



y' 







1.00000 



0.000 



6.000 



1.65 



2.69961 



.30.267 



16.197 



.1 



1.00500 



1.834 



6.030 



1.70 



2.82832 



31.184 



16.970 



.2 



1.02007 



3.669 



6.120 



1.75 



2.96419 



32.101 



17.785 



.3 



1.04534 



5.503 



6.272 



1.80 



3.10747 



33.018 



18.646 



.4 



1.08107 



7.337 



6.486 



1.85 



3.25853 



33.936 



19.551 



.5 



1.12763 



9.172 



6.766 



1.90 



3.41773 



34.853 



20..506 



.6 



1.18547 



11.006 



7.113 



1.95 



3.68548 



35.770 



21.513 



.7 



1.25517 



12.840 



7.531 



2.00 



3.76220 



36.687 



22.573 



.8 



1.33743 



14.675 



8.025 



2.05 



3.94832 



37.604 



23.690 



.9 



1.43309 



16.509 



8.599 



2.10 



4.14431 



38.522 



24.866 



I. 



1.54308 



18.343 



9.258 



2.15 



4..?5067 



39.439 



26.104 



1.05 



1.60379 



19.261 



9.623 



2.20 



4.66791 



40.356 



27.407 



1.10 



1.66852 



20.178 



10.011 



2.25 



4.79657 



41.273 



28.779 



].15 



1.73741 



21.095 



10.424 



2.30 



5.03722 



42.190 



30.223 



1.20 



1.81066 



22.012 



10.864 



2.35 



5.29047 



43.107 



31.743 



1.25 



1.88842 



22.929 



11.3.30 



2.40 



5.65695 



44.025 



33.342 



1.30 



1.97091 



23.847 



11.825 



2.45 



5.83732 



44.942 



35.024 



1.35 



2.05833 



24.764 



12.3.50 



2.50 



6.13229 



45.859 



36.794 



1.40 



2.15090 



25.681 



12.905 



2.55 



6.44259 



46.776 



38.656 



1.45 



2.24884 



26.598 



13.493 



2.60 



6.76901 



47.693 



40.614 



1.50 



2.35241 



27.515 



14.114 



2.65 



7.11234 



48.610 



42.674 



1.65 



2.46186 



28.433 



14.771 



2.70 



7.47347 



49.528 



44.841 



1.60 



2.57746 



29.360 



15.465 







50.000 



46.000 



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 « = 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 : y' = a: a', it follows that aa' y = aif, and y' = a' y^ 

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

 the second by multiplying each by 6. 



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



