NOTES ON STATICS, 647 



In this case, calling the forces F^^ and F^, we have 



F^ = E cos e, F^= R sin 6>, 

 2 (? cos /3 



P = 



R aiQ 29 



iSTow p is least when sin 2d is greatest, — - — being constant. 



R 



that is, when 6* = -, in which case, 



R 



Fr=F,= 



and p 



2GcoB (3 



11. Let Fi and i^2 denote generally the forces which act along the 

 reciprocal lines ; then the volume of the tetrahedron of which these 

 lines are opposite edges 



= iF.F.psin (I _ ^"j 



R cos {9 — ip) R sin Q G cos <p 



cos ip " cos x^i ' R sin 6 ' 



cos 1^ 



= I RG cos je — ip) cos ^ 



cos ;// 

 = i i?G^ cos /3 

 cos /3 cos 



cos (9 — ;//) cos ;p 



And ^ HG cos /3 is constant. "Wherefore, &c, 



12. To find the relative positions of the reciprocal lines and the 

 central axis. 



Let LM, L'N"he the projections of the reciprocal of OA and the 

 central axis on the plane HO A (figure of § 4) ; OM, ON the respec- 

 tive pei^endiculars on these from 0. 



OM 8 sin a • G cos t// sin a G sin a 



Then OL ^-^ ; = p = ^^ :; 7; ■ = ^ ; X • 



cos 1// cos t\i ii sin cos ip R sm 9 



.1 ^T, ON p sin X G sin 3 sin X G sin a 



Also OL^ = __ " — : — ^— — = , 



sin 9 sin 9 R sin 9 R sin 9 



where p is the distance of the central axis from 0, and A the angle 



between the planes BOG, BOA. 



The projections of the central axis and the reciprocal of OA pass, 



therefore, through the same point X at a distance from equal to 



G sin a 



Reind' 



