318 Mr Sang oh the Construction of Oblique Arches. 



X = cot s 



|^v,^.4>.j(ii^)}. 



it is, howeverj more easily detemiined thus 



S X r' + p- tan a. 



r— = cot .' 2 



:: r f tan « 



r= cot s J — cot « + — tan « I 



But S ^ = — J sin ft. S «. whence 



X = _ cot s J ' ■ ~^ sin « -f ^ nep. 1. tan I '^^ + 2 ) l 

 For the end elevation Sve have recourse to equation (P) which gives 

 -^ = ?- J sec « — cos « I and thus 



t/ = ^ J nep, log tan (45 + i «) -- sin « l 



which is the equation of the tractory modified by the existence of the fac- 

 tor -. From this equation the determination of the individual points is 



most easilj' obtained. 



I now proceed to consider the Parabolic Arch. / being the focal dis- 

 tance, the equation of the parabola is 



V- = -if-, whence 11 J u = 2/d z 

 whence again the equation 



«' = cse.|« + -^1 



which belongs to the horizontal projection ; also 



f "^ ) 



.r = cot sj 2 « + TT^r \- 



■ = cot..|v/7{ ^-^7} 



and also 



^ 9/ 

 which arc the equations of t))e three projections. 



I have now run over the equations which serve to determine the dif- 

 ferent parts of oblique, circular, elliptic, and parabolic arches, and had 

 intended to supply examples of the requisite calculations ; but after pro- 

 ceeding to some length in this, it occurred to me that those who have 

 followed the i^receding investigations stand in no need of such illustra- 

 tions, and that these, therefore, woidd merely occupy room without being 

 productive of any benefit, 

 16fA May 1838. 



