Updegraff — Flexure of Telescopes. 259 



Furthermore a rigorous solution of the problem makes neces- 

 sary a solution of the differential equation 



d 2 y 



-t— 2 = — p l x — <fy — axy — bx 7 . 



which takes into account at once the effect of the weight of 

 the tube and that of the object-glass. But on comparing the 

 numerical coefficients of (16) and (37) we see that the 

 astronomical flexure due to the weight of the tube is probably 

 not more than one-tenth of that due to the weight of the 

 object glass and eye-end, supposing that IF = W r The com- 

 puted values, therefore, of the sum of the astronomical 

 flexures due to both causes will not after all be theoretically 

 very uncertain on account of the lack of rigor in the deriva- 

 tion of Eq. (37). 



We shall now consider the displacement of the neutral axis 

 by the compressive and tensile forces. In the above investiga- 

 tion it has been assumed that the neutral axis passes through 

 the center of gravity of each transverse section of the tube 

 and hence that 1 is constant. 



In the case of a telescope tube or beam whose transverse 

 section is constant and symmetrical with reference to a hori- 

 zontal line, the beam being fixed at one end and free at the 

 other and acted upon by a force at the free end perpendicular 

 to the axis of the beam before flexure and in a vertical plane, 

 according to the generally accepted theory the neutral axis 

 passes through the center of gravity of each section.* This 

 results from the assumption (shown by experiment to be 

 approximately correct) that the modulus of elasticity is the 

 same for compression that it is for tension. 



But when the beam is also subject to a force acting parallel 

 to the axis of the beam before flexure, the neutral axis no 

 longer passes through the centers of gravity of the sections of 

 the beam. The amount of this displacement of the neutral 

 axis by the longitudinal force may be computed by the 

 formula, 



* See Weisbach's Mechanics of Engineering, p. 413, Coxe's Translation. 



