444 



Now (' Parabolic Trigonometry,' No. III.) it has been shown that 

 (sec 6 + tan 0) M = sec (6 -- - 1 - to w terms) + tan (0 -- -- to n terms) ; 

 hence in the catenary we shall have 



(y + *)= (y,,,... n +*,... w), 



or (f two points on the catenary be assumed, the abscissa of one being 

 n times that of the other, the nth power of the sum of the ordinate and 

 arc of the latter will be equal to the ordinate and arc of the former. 



We may graphically exhibit with great simplicity the sum of a 

 series of angles added together by the parabolic or logarithmic 

 plus J -. 



Let a set of equidistant ordinates for simplicity let the common 

 interval be unity meet the catenary in the points b, c, d, k, I, and 

 then let the catenary be supposed to be stretched along the hori- 

 zontal tangent passing through the vertex A. Let the points 

 b, c, d, k, I, on the catenary in its free position, coincide with the 

 points (3, y, S, ic, X, on the horizontal line when strained in that posi- 

 tion, and as x or A/3, Ay, A 2, Ac, AX is successively equal to 



m, 2m, 3m, 4m, &c. or to 1, 2, 3, 4, &c. if i=l, 

 we shall have 



2y =e 1 + e~ 1 s =e l e~ l 



2y, = 



Hence the angle AO/3 or e is such that 



sec 6 + tan e= e, 



AOy such that, sec AOy + tan AOy = e 2 , or AOy=e-i- 

 AOS such that, sec AOS + tan AOd = e 3 , or AOS = e -*- s ->- e ; 

 consequently if we draw lines from the focus O successively to the 

 points ft, y, g, K, X, the angles AO/3, AOy, AO3, AO*, AOX will be 

 the angles s, e-^-e, s- L e- L s, e- J -e- 1 -e- 1 -e. 



This is one of the simplest graphical representations we can have 

 of angles added together by the parabolic or logarithmic plus - 1 -. 



Hence as successive multiples of an arc of a circle give successive 

 arithmetical multiples of the corresponding angle at the centre, so 

 successive multiples of a given abscissa give successive arcs of the 



