052 DR WALLACE ON A FUNCTIONAL EQUATION. 



It is easy to see how, from the formula, an approximate geometrical deter- 

 mimition of points in the catenary may be obtained. 



45. It has been found that, x denoting the amplitude of any point in a ca- 

 tenary, ?/ the ordinate at that point, and (p the angle which a hne touching the 

 curve at the top of the ordinate makes with the horizontal axis, then (Art. 43), 



dx=asec(p . d<p, 

 and z = a/ sec (p . d(p . 



Suppose sec 4> to be expressed, not in decimal parts of the radius, as in the 

 common trigonometrical table, but in units, each of which is the arc that mea- 

 sures an angle of one minute of a degree ; of these, the radius contains 3437.74677. 

 Let n denote this number, and suppose d(p to he one of these units. The inte- 

 gral/sec cpcl4> will be approximatively expressed by the series 



— f sec r + sec 2 + sec 3' + sec 4' + &c.} 

 n 



and x=— {secr + sec2' + sec3' + sec4' + &c.} 



n 



Now the sum of the series continued to as many terms as there are minutes in 

 the angle <^, is known to express the length of the enlarged meridian in Weight's, 

 or as it is called (improperly) Mercator's projection of the sphere ; and these 

 sums are o-iven in nautical tables under the name of meridional parts, therefore, 

 puttino' M {(p) to denote the meridional pai-ts of a latitude (p, and this angle (p be- 

 ing found from either of these formulae, 



, f{x) ^ , F(z) , Mcp 



sec(i) = ^^-^^, tand) = — ^-^ ; we have x=a . — ^ ; 



Or we may first find (p, and then/(^r) and F (x), from these formulae, 



(1) U{<P) = !^; 



(2) /(.r) = 8sec^ = — ^- ' (^^) 

 ^ ' -^ ^ ' ^ cos (/) 



(3) Y(x)=aia.n<p. 

 Example. Let the parameter of a catenary be 100 feet ; it is proposed to find 



the ordinate /(.t) and the axe F {x) to the amplitude a; =125 feet. 



Log. Log. Log. 



«= 3437.7 3.53627 a 2.00000 tan<^ 10.20477 

 a; =12.5 2.00691 cos^ 9.72381 a 2.0000 

 a=100 Ar. comp. 8.00000 /(ar) =188.88 2.27619 F (a-) = 160.24 2.20477 

 M((^) = 4297.1 3.G3318 

 (^=58' 2'. 

 Here we first find M ((/>) to be 4297.1, which, by inspection in a table of me- 

 ridional parts,* gives ^ = 58° 2'. The angle (p being known, f{x) and F [x) are 



• Mrndoza llios' Collection of Tables pr Navigalinn ; or any treatise on navigation. 



