Mr Sang* on the Construe Hon of Oblique Atckoi. SIS' 



Lastly, we have 



"ixi . ))n 111 1 z ^^ 



:^ = sin S -r r— = tan I = -r- . . . (P) 



iz Iz iz 6U 



■whence it is, that the end elevation of the joint crosses that of the line of 

 pressure at right angles. 



Before proceeding to apply the above differential equations to particu- 

 lar cases, the following recapitulation may be made : 

 Equation H gives the Horizontal Projection. 

 L ... Developement. 



... Side Elevation. 



P ... End Elevation of the Joint. 



And it is to be remarked, that these equations are absolutely general, 

 applying to every skewed cylindroid arch. 



Having now completed the general investigation, I proceed to apply 

 the principles to specific cases ; in the first case to the circular arch. 

 Denoting by r the radius of the circle, we have 



I := — , ; =: r cos — , M = r sm —:•■' + u^ = r' ; 

 r r r 



equations which take the place of (B) in the general analysis. 

 For the horizontal projection of a joint we have 



-r— = sm 



0°^0'= 



and thus 



= cse s. 



d U 



•whence integrating 



r = r. cse s. nep. log 



^/(:^) 



Now v' = r. cse s. nep. log (r -f «) is the equation of a logarithmic curve 

 to oblique co-ordinates having one side of the semicylinder for its axis, 

 and r cse *. for its subtangent : while — v" =: r. cse s. nep. log (r — u) is 

 that of a similar curve having the other side of the semi-cylinder for its 

 asymptote, and thus the v of the joint which is the arithmetical mean of 

 these is obtained by bisecting the interval between the two logarithmics. 

 Passing to common logaritlims, and putting M for the modulus, 

 .43429448, etc. we have 



r. cse »,»• + « 



»M» 

 lOrce. _1 



•i M r 



The horizontal projection of the joint of a circular skewed arch Is thus 

 a new curve to wliicli I have given the name of Double Logaritlimic : the 



