652 T^^ WALLACE ON A FUNCTIONAL EQUATION. 



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

 mination of points in the catenary may be obtained. 



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

 tenary, y the ordinate at that point, and <p the angle which a hue touching the 

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



dx=asec<p . d<p, 

 and x=a/Bec(f) . d (p . 



Suppose sec <^ 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 ^ to be one of these units. The inte- 

 gral/* sec (pd^p will be approximatively expressed by the series 



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

 n 



and «= — { sec r + sec 2' + sec 3' + sec 4' + &;c.} 



n 



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

 the angle <P, is known to express the length of the enlarged meridian in Wright's, 

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

 sums are given in nautical tables under the name of meridional parts, therefore, 

 putting M {(p) to denote the meridional parts of a latitude cp, and this angle cp be- 

 ing found from either of these formulae, 



sec 9=^^-^^, tan 0= — ^^ • we have x=a . — ^ ; 

 Or we may first find (p, and ihen/{^) and F (.r), from these formulae, 



(1) M(CP) = !!±; 



(2) /(:.) = « sec = -^; f (1^) 



cos 9 



(3) F (x) = atan(p. 



Example. Let the parameter of a catenary be 1 00 feet ; it is proposed to find 

 the ordinate /'(c^') and the arc F {w) to the amplitude x—l2o feet. 



Log. 



«= 3437.7 3.53627 

 z = 125 2.09691 



Log. 



a 2.00000 

 cos (p 9.72381 



Log. 

 tan (p 10.20477 

 a 2.00000 



a = 100 Ar. comp. 8.00000 

 (0) = 4297.1 3.63318 

 0=58° 2'. 



/(a:) = 188.88 2.27619 



F (a;) = 160.24 2.20477 



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

 ridional parts,* gives 0=58° 2'. The angle (p being known, f{x) and F {x) are 



* Mendoza Rigs' Collection of Tables for Navigation ; or any treatise on navigation. 



