252 



Trans. Acad. Sci. of St. Louis. 



Fig. 4. 



which is the formula for the deflection of a horizontal beam 

 fixed at one end and loaded at the other.* 



We now pass to the case of flexure due to the weight of the 

 tube of the telescope. Let the weight per unit of length be 



to. Then, the origin of co-ordinates 

 being taken, as before, in one case at 

 the object-end and in the other case 

 Y at the eye-end of the telescope, we 

 have as the weight on any section, 

 ivx. The components perpendicular 

 and parallel to the line of collima- 

 tion will be wx sin# and iox cos#, d 

 being as before the angle which the 

 line of collimation in its undis- 

 turbed position makes with the 

 vertical. In Figure 4 let OX and 

 OY be the axes of co-ordinates for the upper half of the tube. 

 Evidently the moment on any transverse section, as that pass- 

 ing through the point s, will be ~wx 2 s'\nd very nearly. We 

 here assume sO to be a straight line, which when the deflec- 

 tion is small may be done without appreciable error. The 

 case of the longitudinal component is however less simple, 

 since in this case, except for sections near 0, we cannot assume 

 sO to be a straight line. If we assume the curve CsO to 

 be an arc of a circle, the lever arm of the compressive force 

 for any section would be \y, while on the assumption of a 

 straight line the lever arm for any section would be \y. 

 The problem seems not to admit of direct and rigorous solu- 

 tion since a knowledge of the equation of the curve CsO is 

 needed in forming its differential equation. Evidently, 

 however, the mean value of the lever arm of the compress- 

 ive force is somewhere between \y and \y, and we adopt ^y. 

 This gives as the moment of the longitudinal component of 

 the weight on any section ^ivxy cosd. We shall see later 

 that whatever error there may be in this assumption will not 

 for our purpose seriously affect the result. 



We then get as the approximate differential equations of 



See Wood's Resistance of Materials, p. 109. 



