284 SUPPLEMENT TO CHAP. XIV. [CHAP. n. 



have in view, and we shall not, therefore, pursue it further. We have 

 already all the general formulae of which we shall need to make use in the 

 discussion of the parabolic arch. 



6. Circular Arc concentrated Load. In a perfectly similar 

 manner we may make out analogous formulae for the circular arch. Thus, 

 referring to equation (8), Art. 1, and inserting for Q and M their values as 

 given in (18) and (20), Art. 3, we have for the force in the direction of 

 the axis (see eq. 4), 



. Ads 



E A = H cos o V sin i 

 ds 



- Ad*' 

 1 A = H cos a V sin a 



ds 



Putting for H, Vand V their values from (21), Art. 4, and (17), Art. 3, we 

 have, 



T! A A ^ * .p sin a + sin /3 2 cos a sin 



~Ta ' 2(1 -cos a) 



_ . A d a' _ _ sin a sin /3 

 ds 2(1 cos a) 



(a) Change of direction of tangents. 

 Referring to equation (12), Art. 1, we have, since rd<f> = ds and r<f> = t, 



!*t-T?* 



The value of M is given in (20), of in (24). Inserting these values, 

 we have 



A <f> = j=-= / H (cos <fr cos a) V (sin a sin 0) \d (f> 



1 

 =TA~ C 3 CO8 a + V sin a) 0. 



Performing the integration,* and putting, for brevity, K = - ,, we have for 



Ar 



the three segments of the arc, as before, 



H (sin <f> <i> cos a) V (< sin a + oos <j>)\ K r a (H cos a + V sin a) + A ~) 



: 



I K r s (H cos a + V sin a) A + A' V 26 



For the point E, $ = /3, and the first and second equations become simul- 

 taneous. Hence, after reduction, 



A - A' = (V - V) r sin a + (V + V) r* cos + K (V- V) r sin a. 



But V + V = P, and from (17) V V = P ^-^, hence 



sin a 



/M 

 si- 



A A' = Pr z cos + (1 +*)/3sin/3 I . . . (I.) 

 * See Art. 7, following. 



