Mathematical and Physical Papers. . 225 



whence, using the author's formula, 



Ic = ^^' -. 7854 X (5.87)2 = 1365 -27 -1338 

 4-y-8-7.37-.63" 



Our added term is then bhf — — y ) =64X'632=25, which shows 



that the missing term is, in certain cases at least, about as important 

 as the second term of the formula given. 



The next formula to which we direct attention is formula {15). 

 We will apply the second method used in deducing formula (1). 

 Now, however, we shall take moments about the top of the beam 

 instead of the bottom. 



Ae^ = the area to be added to take the excess of stress in steel in 

 compression = ( e — 1 ) Ag^. 



Ae = the area to be added to replace the steel in tension —eAg. 



eA3+(h"+dii) + (e-l)AsW+^=x[eA3+(e-l)A3Vbx] 



2 2[eAs + (e— l)Asi3x ^ 2eAs(h"+dii)+2(e-l)As'dii 

 ^+ b ~ b 



whence 



e(As4-Asi)-Asi 

 x= — : 



■4 



. [e(Ae + As')-As^]2 2e[As(h" + dii) +As^dit] -2Asidii 

 ^ b2 b 



The difPerence here is not large unless Ag^ is large. And generally 

 if the concrete did not take tension, all the reenforcing would be 

 on the lower or tensile side of the beam. But in formula (1) the 

 author multiplies his areas by (e — 1) ; so, to be consistent, he should 

 do the same here for the area in compression. 



We have not discovered any further errors in this set of formu- 

 las. There is, however, an important error in their application to 

 which we wish to call attention. The author uses this example : 



"What is the safe bending moment for a reenforced concrete 

 beam 16 inches deep, 4 inches wide, having one steel rod f inch 

 in diameter on the tensile side only, so placed that the center of 

 -15 



