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III. "On the Application of Parabolic Trigonometry to the 

 Investigation of the Properties of the Common Catenary/' 

 By the Rev. JAMES BOOTH, LL.D., F.R.S. Received 

 March 19, 1857. 



Some time ago, on the publication of a paper read by me last 

 summer at Cheltenham before the Mathematical Section of the 

 British Association on Parabolic Trigonometry and the Geometrical 

 origin of Logarithms, Sir John Herschel called my attention to the 

 analogy which exists between the equation of the common catenary 

 referred to rectangular coordinates, and one of the principal formula? 

 of parabolic trigonometry. Since that time I have partially investi- 

 gated the subject, and find, on a very cursory examination, that the 

 most curious analogies exist between the properties of the parabola 

 and those of the catenary, that in general for every property of the 

 former a corresponding one may be discovered for the latter. In 

 this paper I cannot do more than give a mere outline of these 

 investigations, but* I hope at some future time, when less occu- 

 pied with other avocations than at present, I may be permitted to 

 resume the subject. I will only add, that the properties of this 

 curve appear to be as inexhaustible as those of the circle or any 

 other conic section. 



II. The equations of the common catenary referred to rectangular 

 coordinates are 



The point O may be called the focus, whose distance to the vertex A 

 of the curve is =m. 



It will simplify the investigations, without lessening their gene- 

 rality, if we assume the modulus or focal distance m 1 . 



Assume 2secd=e*+e~ x , 2tand=e x e~ x (1.) 



Theny=sec0, s=tan0 (2.) 



Now if we make x', a", x'", &c. successively equal to 2x, 3x, 4x, 

 &c., we shall have (see ' Parabolic Trigonometry,' No. XXVI.) 

 y =sec s =tan 6 



y' = sec (0-"-0) s' = tan(0->-0) 



y"= sec (0J- 0-1-0) s" = tan (0-L 0-i-0) 



3/"= sec (0-i-0-!-0-i-0) *'"= tan (0-L 0-i-0-i-0) 



VOL. VIII. 2 L 



