446 



3D, K-K, XL to the parabola, the differences between these tangents 

 and the corresponding parabolic arcs, namely, AD, AC, AD, AK, 

 AL, will be m, 2m, 3m, 4m, or 



AB-/3B=m, AC-yC=2m, AD-SD=3m, AK-*K=4m &c. 

 This is evident (see ' Parabolic Trigonometry,' No. XXVI.) for the 

 angles AO/3=, AOy=e- L -e, AO2=e-J-e-i-, AO(c=e- 1 -6- J -e J -e. 



We may further extend these properties of the catenary. To sim- 

 plify the expressions, let Y^> denote the ordinate of a point on the 

 catenary at which the tangent makes the angle with the axis of X. 

 Let S0 denote the arc measured from the lowest point, and let X< 

 signify the ordinate. 



Then S0= tan 0, Y0= sec f 



Now let x, x t , x tl be the abscissae of the three arcs whose tangents 

 make the angles $, x , w with the axis of x, and let the equation of 

 condition be simply 



Then we shall have the following relations between the corresponding 

 arcs and ordinates of the catenary 



when x t =x 



2S 2 0=Yo,-l 



Let there be four arcs of the catenary whose abscissae x, x t> x,,, x ttt 

 shall be connected by the following relation 



XIH-BH + BH or a?,,, = 



Let w, <f>, x , ip be the corresponding angles made by the tangents 

 to the extremities of the arcs Sw, 89, S x , Si//. 



Then we shall have the following relations between the arcs and 

 the ordinates 



Hence also 



