64 STRENGTH OF. MATERIALS 



If the point B lies on the ellipse + g = 1, if coordinates must 

 satisfy this equation, and, consequently, 



* _i 



(32) + F~ 



In this case the neutral axis passes through a point on the ellipse 



diametrically opposite to B; for if -J, 



and z in equation (31), it is evident that the condit 



The tangent to the ellipse |+ = 1 at the point - 

 z ^^y]L--\ which is identical with equation (31). Consequently, 



~2 ~ J2 



if B lies on the inertia ellipse, the neutral axis corresponding 

 tangent to the ellipse at the point diametrically opposite to B. 



From equation (31), the slope of the tan- 

 gent is found to be 



If, then, the point B moves out along a radius 

 CB,z* and / increase in the same ratio, and 

 consequently the slope is constant : that is to 

 say, if B moves out along a radius, th,. n , Mi- 

 tral axis moves parallel to itself. 



As d and y 1 increase, z and y must de- 

 crease, for the products zz 1 and yy' mu>i In- 

 constant in order to satisfy equation (31). 

 FlG - 48 Therefore the farther /; bi tnm the < 



gravity, the nearer the corresponding neutral axis is to the c 

 of gravity, and vice versa. 



If, in Fig. 48, TN is the neutral axis corresponding to H. it 

 lows, from the above, that CB CT is a constant wherever B is on t In- 

 line BT. But if B lies on the ellipse, the corresponding neutral axis 

 is tangent to the ellipse at the point diametrically opposite to B t and 

 in this case the above product becomes CM . There! 



(33) CB : CT=CM*. 



From this relation, the position of the neutral axis can be determined 

 when the position of the point B is given. 



