314 Mr Sang (fii the Construction of CMique Arehea. 



1 II . d «* 



I v ~ '' d u' + d i^ 



in which the characteristic S refers to the joint, d to the line of pressure. 



But r-z — r"5" is the square of the cosine of the inclination of the line of 

 du- + dr ^ 



pressure to the horison ; whence, if we denote that inclination by i, 



ou 



-;:— = Sin .<!. cos j2 . . . (H) 

 or 



When, then, as is the case at the crown of the arch, i is zero, - — =: sin s ; 



d V 



J« III 5 !/ 



but -r- := — i— + sin s, so that, at the crown, -^ =:. o, that is, the 



horizontal projection of the joint, is there perpendicular to the parapet, 



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



ly . . , . 

 decreases, and therefore ^r- =■ sin s. sin i' (I) must increase : 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 abutment. 



Sr 

 Smce — =: sec s, the above equation put in rectangular co-ordinates 



becomes, 



~ = tan s. sm i= . . . (K) 



iV 



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



: ~r and equation H becomes, 

 da 



and thus, if we imagine two joints running quite close to each other, cut- 

 ting the crown-line at the minute distance S v, the distance S a, intercepted 

 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 



i a 



^— = tan .<■•. cos ( . . . (M) 



Again, we have cot»; whence equation H becomes, 



c 



-T— = sm s. sin i. cos i = J sin s. sin 2 i. . . (N) 



I' 



— = tan .<. sin i. cos i = i tan ». 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. 



