[546] 

 NOTES ON STATICS. 



BY JAMES LOUDON, M.A., 



MATHBHATICAL TUTOR AND DEAN IN UNIVEBSITT COLLEOK, TORONTO. 



The following notes are in continuation of what was published in 

 the Canadian Journal for February, 1872. All these proofs were 

 devised by the writer during the year 1868, but were not communi- 

 cated to the Canadian Institute until November, 1870. 



7. On transferring the origin from to a point 0' at a distance r 

 on OA, we will have there E, G, and Rr sin 6, the axis of the latter 

 being parallel to OD, perpendicular to the plane ROA. The axis of 

 the resultant couple G' at 0' is therefore in a plane parallel to DOC, 

 and the plane of G' must therefore meet the plane of G along a line 

 parallel to OB. 



8. If OC be perpendicular to DO A, the plane of G' will intersect 

 BOA along OA, in which case the reciprocal lines will coincide or be 

 parallel. 



9. To find the moment centre of a plane passing through a given 

 line. Let the plane be the plane of the paper, OA the given line, 

 R, G, the force and couple Sit ; a the angle which the axis of G 

 makes with the given plane ; OF in the given plane perpendicular to 

 the axis of G ; OD perpendicular to ROF ; y the angle ROF ; d the 

 angle between OD and the given plane. 



Then on transferring to 0' at a distance r in OF, we have there R, 

 G, and Rr sin y, the axis of the latter being parallel to the plane 

 GOD. Therefore the axis of G', the Resultant of G and Rr sin y, 

 will be perpendicular to the given plane if 



G cos a = Rr sin y cos d, 



G cos a 



or r = ^— . . 



Ji sin 7 cos 



which determines the position of 0', the moment-centre of the given 



plane. 



10. The angle between OA and its reciprocal is plainly 2 — ^' 



The lines are therefore perpendicular to each other when ^ = 0, 

 which requires OA to be coincident with OC, and therefore 

 cos = cos 6 eos a. 



