DR WALLACE ON A FUNCTIONAL EQUATION. 657 



F the focus : Let a straight line NE touch the curve at any point N, and 

 let FE be a perpendicular from the focus on this line : Let z denote the 

 parabohc arc VN, t the segment NE of the touching line between the point 

 of contact and perpendicular, and p the perpendicular : Let ABC be a ca- 

 tenary, of which CQ is the horizontal axis, and BC = a the parameter, 

 which is equal to FV, one-fourth of the parameter of the parabola : Let 

 CQ, = X, and PQ =/{x) be co-ordinates at any point P of the catenary : 

 The parabola and catenary are so related, that if ,v=z—t, then/(^) =p. 



52. Suspended bridges are now very common ; and there is a species of bridge 

 coming into use, the arch of which is convex upward, and formed by uniting several 

 bended planks with oak trenails ; this kind of bridge is, in some places, carried 

 across ravines in the line of railways. I know not whether engineers erect these 

 upon the principle of equilibrium, but I believe it quite possible that such arches 

 may advantageously have the form of curves of equilibration, with straight road- 

 ways. 



53. The construction of a catenary, also a curve of equilibration, must be 

 greatly facilitated by a table of co-ordinates of a catenary ; and I have already 

 stated, that such a table has been actually given by the late Davies Gilbert, 

 Esq.* The formulae of this memoir give great facilities for the construction of 

 such tables, and I have computed those here given by the following formulae. 



Continuing the notation of art. 39, and assuming the parameter a to be = 1, 

 we have found 



These expressions, by development, give 



/(^) = l+y:2^ 1.2.3.4 + 1.2.3.4.5.6 ^^" 



^^"^=" + t4:3^ 1.2.3.4. 5 •^ 1.2.3.15. 6.7 "^ ^^- 

 Some of the numbers in the tables were found by these series, as 



^w=i+t^^t:to-^^°- '^W'^^m^ 1.2.3. 4.5 + ^°- 



•^W=l+ 10^+ 10.20^30.40 + fa- ^«=Rr+io^r3o + *"- 



■^(•'*»i)=^+ ioooiTioooo ^ ""■ ''(■«"'i)= vm>* *=«• 



* Philosophical Transactions for 1826, Part iii. I have been told that the very ingenious author 

 of this memoir did not himself compute the numbers, which are almost all incorrect. 

 VOL. XIV. PART II. 6 H 



