NOTES ON STATICS. ' 233 



Therefore the above line may also be deterniiaed by the intersection 

 of the momental-plane at with a plane parallel to the plane of the 



(r cos 



paper at a distance = — t—t • 

 ^ ^ ^ sm 



The line so determined and OA are said to be reciprocal to one 



another. 



4. When R oi and the couple G are replaced by two forces, one 

 of which acts along the line OA, to find the magnitude and line of 

 action of the other. 



Let the forces of G be S acting along OB, and >S^ at a distance -^ , 



the figure being the same as in previous proposition. 



Now if the resultant of R and S at acts along OA, we must have 



R sin = S cos 4' 



RsinQ 



cos ■^ ' 



and therefore the other force acts at a distance = — , that is, along 



i? sin ' ' " 



the reciprocal of OA. 



The resultant of R and >S at = R cos -\- S &\n (p 



R cos (9 — •^) 



cos ^ 

 Therefore the two forces are 



^i^iiini) alo„, OA, 



cos "- 



- ^ sin . . 



ana along its reciprocal. 



cos ;|/ ^ 



5. If /S be the angle between the axis of G and the direction of R, 

 the values of the forces may be written 



^^^ along OA, 



COS0 ° 



and — — {cos^^ -)- cos^/? — 2 cos /? cos ^cos cc}i" along its reciprocal. 



6. The shortest distance between the reciprocal lines is evidently p 

 already found in § 4 to be 



R sia 0' 



November 12, 18*70. 



(To be continv-ed.J 



