196 MEMOIR OF THE LATE PROF. E. HOUGKINSON, F.R.S. 



A B be the fixed supports of the beam^ A D B its position 

 after the weight has been placed upon it. If the point E 

 in the beam is sufficiently near to D;, then the line D E 

 may be considered straight ; produce D E to meet the hori- 

 zontal line A B in F, and put the angle D F B = ^. Let 

 the force H^ applied to move the weight from D to E upon 

 the inclined plane D E, be in the direction of D E. It is 

 evident that the force H can be decomposed into two, viz., 

 H sin 6 acting vertically upwards, and H cos acting hori- 

 zontally from D to E. If we examine the effect of these 

 two decomposed forces, it will be found that the force 

 H cos 6 J which is nearly equal to H, since the angle 6 is 

 small, will produce an elongation in the beam A D, and a 

 compression in the beam D B. When the elongation of 

 A D is greater than the compression of D B, the beam 

 between the supports is increased in length, hence the 

 middle point D, where the weight is placed, is moved verti- 

 cally as well as horizontally into another position. From 

 this force alone the beam would become a wreck if the force 

 H, or the velocity with which the weight is moved from 

 D to E, was sufficiently large ; but, to prevent this catas- 

 trophe, the vertical component force H sin 6 diminishes 

 the reaction of the weight and beam. The vertical force 

 of the weight, instead of being the weight alone, is now 

 diminished by H sin 6, and is become W — H sin 9, where 

 W is the weight of the moveable body. The effect then of 

 the vertical component is directly opposite to that of the 

 horizontal component ; and it is evident that under certain 

 conditions either one or the other of these two forces may 

 prevail. Hence the indications of theory are in harmony 

 with the observations of engineers, and fully justify the 

 conflicting evidence which they have given on the subject. 

 Sometimes the conditions of the moving weight and the 

 beam are such as to produce a statical deflection greater 

 than the dynamical, and sometimes the conditions produce 



