THE EQUILIBRIUM OF THE CIRCULAR-ARC BOW-GIRDER. A415 
while the upward deflection at P, due to P, 
3 { 0845 , 0085 ) 
=Pr? ET t acy fm (13) and fig. 5, 
and the upward deflection at P, due to force Q 
au cezon Old 
=Qr{ Ser t acy \ from (13) and fig. 5. 
Collecting and equating deflections at the same points, gives :— 
wr(-7272CJ + -053ET) = Q (-4674CJ + 0382ET) + P(-4220CJ + :0594ET), 
wr(:39280F + -0213EI1) =Q (-2297CJ + O15EI) + P(-2710CJ + 0140EI). 
If El=1°24CJ, the solution of this gives— 
="T4dwr; P='83ur. 
From this 
R,=R, =4{zwr - Q - 2P} 
='37ur. 
Also 
M, + M, = 2wr? — 2Pr sin 45° — Qr 
= '088wr? 
.. M,=M,=:044ur? 
while T, (from figs. 5 and 7) 
‘298wr2 —-114Pr —+182Qr — -081 Pr 
(298 — +297) wr? 
= 0010 wr’. 
§ 9. APPLICABILITY OF ForRMUL& To SECTIONS OTHER THAN CIRCULAR. 
Although the formule have only been checked by experiments on beams of circular 
section, they should apply with equal accuracy to any section of which the EI and CJ 
are known. ‘The former of these products is usually known with some close degree of 
approximation, for any commercial section. While the value of J may also may also be 
readily calculated, the value of C is not known with nearly the same degree of accuracy 
as is that of E. Moreover, the product CJ as used in the simple theory of elastic torsion 
is only constant for different values of 6, for a circular section. Its value for such 
sections as are usual in girder work, cannot as yet be predicted with any great pretensions 
to accuracy. It is hoped to carry out an experimental investigation on this point at an 
early date. 
Owing, however, to the fact that the maximum torque is in general much less than 
the maximum bending moment, the resultant stresses on the girder, at the point 
where this is most heavily loaded, are only slightly affected by even a considerable 
variation in the value of CJ, and for all practical purposes the constants given in this 
paper may be taken as applying to any normal section. 
TRANS. ROY. SOC. EDIN., VOL. XLVIII. PART II. (NO. 20). 62 
