445 



catenary which, extended along the vertical tangent, subtend at O 

 successive parabolic multiples of the original angle. 



Since the original interval was assumed equal to m, and as the arc 



i 



J K. 



of the catenary is always longer than the abscissa which subtends it, 

 or A/3>A6, it follows, as has been shown in * Parabolic Trigono- 

 metry,' No. XXII., that g>45. 



Since ^=1 (*-*-) we shall have ^=tan 0, but as ^ is the 

 dx 2 ^ dx dx 



trigonometrical tangent of the angle which the linear tangent at the 

 point {xy) makes with the axis of the abscissa, hence this other 

 theorem : 



Let a set of equidistant ordinates meet the catenary in the 

 points b, c, d, k, 1, and at these points let tangents to the curve be 

 drawn, they will be inclined to the axis of the abscissa by the angles 

 6, 0-1-0, 0-J-0-J-0, 0- L 6- 1 -0-L0, &c., which is even a yet simpler geo- 

 metrical representation than the preceding. 



Hence also it evidently follows that as the limit of the angle which 

 a tangent to the catenary makes with the axis of the abscissa is a 

 right angle, the limit of the angle d-*-d- i -d- L -d- L -d, ad infinitum, is a 

 right angle. 



We have also this other theorem : 



If with the point O as focus, and A as vertex, we describe a 

 parabola, and from the points fi, y, S, K, \ we draw tangents /3B, yC, 



