1840] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



23S 



or „.'..= ./(-^?li^\ 

 /y/ l sin s — e' vj 



(*' z)- 

 or 6' « = sin s _, , ^ , — - 



The last formula may also be put thus : 



(G) 



— • = sm s , „ , , J 

 5 a u- ■\- a z' 



in whicli the characteristic S refers to the joint, d to the line of pres- 

 sure. But -; "-, — is the square of the cosine of the inclination 



d u- -\- d :- 



of the line of pressure to the horizon ; whence, if we denote that in- 

 clination by /, 



— := sin 8. cos r . . . (Hj 

 5 V 



S II 

 When, then, as is the case at the crown of the arch, ; is zero ^-^ = 



sin s ; but - = + sin s so that, at the crown, . - = o, that is, 



the horizontal projection of the joint, is tliere perpendicular to the 



parapet, as might easily have been anticipated ; but when i increases, 



5w 

 its cosine decreases, and therefore -=i- = sin s. sin »' (I) must increase: 



So ^ , 



that is, the line must bend away, from being perpendicular to the 

 parapet, until, if i could reach 90°, it would be parallel to the abut- 

 ment. 



Since ^^ ^ sec s, the above quotation put in rectangular co-ordinates 



S.V 



becomes. 





tan s. siu i'^ 



(K) 



If a he taken to represent the arc of which u is the projection, cos i 

 du 



= -T- and equation H becomes, 



Sa 



5 II 



sm s. cos 2 



.(L) 



and thus, if we imagine two joints ruiming quite close to each other, 

 cutting the crown-line at the minute distance 5 r, the distance Sa, in- 

 tercepted between them on the arc, or the breadth of the course, is 

 proportional to cosine i. 

 The above equation can also be put under the form 



So 



-- =: tan s. cos i 



■ (M) 



St{ 



Again, we have g . = cot i; whence equation H becomes, 

 Sc 

 So 

 iz 



:;= sm s. sin 2. cos i ::^ * sin s. sin Z i. 



(N) 



S.v' 



. tan s. sin j. cos 2 = 4 tan s. sin 2 i. 



. . (O) 



From which it will be seen, that the general statement made as to 

 the side elevation of the joint is true. 

 Lastly, we have 



5 )/ , Sv Sii 



r^ = sin S r — : 



Sr Sj Sz 



Sz 



: tan 8 = ^ 



oil, 



' (P) 



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

 of pressure at right angles. 



Before proceeding to apply t)ie above differential equations to par- 

 ticular cases, the following recapitulation may be made : 

 Equation H gives the Horizontal Projection. 

 L . . Development. 

 O . . Side Elevation. 

 P . . End Elevation of the Joint. 

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

 apijlying to every skewed eylindroid 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 / the radius of tlie circle, we have 



a a 



i =: -, « = )• cos -, tc : 



r r 



r sin - ; z- -\- li^ ■=. r'' ; 



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



For the horizontal projection of a joint we have 

 Sm_ 

 5 y 



(^■^os "^ i= sin 



and thus 



So 



J- = cse «. 



whence integrating 



t. = r. cse 8 nep. log ^ {jJ^^ 



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

 curve to oblique co-ordinates having one side of the semicylinder for 

 its axis, and r cse s. for its subtaiigent: while — ti" = r. cse s. nep. 

 log (;• — u) is tliat of a similar curve having the other side of the 

 semi-cylinder for its asymptote, and thus the ji of the joint which is 

 the arithmetical mean of these "is obtained by bisecting the interval 

 between the two logarithraics. 



Passing to common logarithms, and putting M for the modulus. 

 •43429ilS, &:c. we have 



r. cse s , »• + a 



2 M V 

 10 ;• cse 8^1 



tizzzr 2 M P • 

 10 r cse s + 1 



The horizontal projection of the joint of a circular skewed arch is 

 thus a new curve, to whicli I have given the name of Double Loga- 

 rithmic : the analogy between this curve and the common catenary has 

 already been pointed out. 



In order to trace the side elevation, we must resume equation (O) 

 which, when adapted to the circular arch, is 



^ = tan8.^^(^) 



whence 



;•. cot s 



nep. log 



r -\- y' j-2 — s- 



r — Vjj ,3 



= nep log 10. r. cot s log tan { 45^ + ,~ ) 



But the equation 



^' = ^ nep. log . —Vi- — I- 



~ r — V r — z 



is just the equation of the tractory, whence 



whence 



is the equation of a curve having its ordinates greater than those of 

 the tractory by the quantity V*"' — -•', this curve I have named the 

 companion lo the traclonj, or, on account of the connection which is ex- 

 plained in the paper, and which at once flows from the above, the in- 

 i-erttd ca/eiiary. 



The equation for the end elevation of a joint adapted to the circular 

 arch is 



which is the well known equation of the tractory. This is the cliarac- 

 teristic curve of the circular oblique arch : as all tractories are similar 

 to each other, it is easy to make a table of its co-ordinates. 



The preceding equations enable us to obtain any one of the projec- 

 tions of the joint, and are essential to a knowledge of the nature of the 

 different curves. They are, however, inconvenient when we wish to 

 ascertain the dimensions of the individual arch-stones, and need, for 

 that purpose, to know the intersection of the joint with any one of the 

 lines of pressure. The equation of the development furnishes us with 

 the means of obtaining these points, as well as all the projections, by 

 processes remarkable for their simplicity. To find this equation I re- 

 sume (L) which, adapted to the circular arch, becomes 



2 I 2 



