A Treatise on the Theory of Screws

n 302 THE THEORY OF SCREWS. [283, multiplying the equations severally by Pi,p2, ••• and adding, we get å tp^ »i /Si = I y"'pv (A cos (yi) + ß2 cos (772 ) + ... + A cos (yf>)) = i y"'Pri cos (ßy) = y"™^ (since pv is indefinitely large), whence we proceed as before and we see that the theorem (iii) remains true, even if pv or p^ or p^ be infinite. If pa be zero, then in general cos aa> is zero. But in this case pa 4- cos am becomes da the length of the perpendicular from the centre of gravity upon a. Hence we have 7>i«i PM p^ da. da da and the proof proceeds as before so that in this case also the theorem holds good. Finally, letpabe infinite, a> must then be of zero pitch and pass through the centre of gravity and å-po. = a> We have &>! = J cos(al), a>:! = | cos (a3), ®5 = cos (as), ... so that the equations (i) become i w" cos (ai) = y"^ + p"'pi, I to’" cos (a3 ) = y"'y3 + p'"p3, I co"' cos («5) = y"'y:, + p'"p6, |<ow cos (ai) = y'"y-2 + p'"p2, % a>" cos («3) = y"yi + p"pt> l<o"' cos (as) = y"'y6 + p"'p6- Multiplying these equations by + aßlt — aß2, + bß3, —bß3,... and adding, we have 2 w'" [a (/Sj - ß.2) cos (ai) + b (ß3 - ß±) cos («3) + c (fis - ße) cos (as )] = y"'^ ■ Let a be the screw belonging to the reciprocal system on which there is an impulsive wrench of intensity a'" due to the reactions when an impulsive wrench is administered on Then we have ßaßi = ; — ßaß,2 = ^'"^.2 + a"'a-.2, whence ßa (ß2 — ß.2~) cos (ai) = cos (£1) cos (ai) + a"' cos (ai) cos (al), with similar expressions for the two other pairs, whence by addition ß [a - &) cos (ai) + b (ß3 - /34) cos (as) + c (ß5 — ßß cos (a 5)} = cos (af), for since a and <r are reciprocal and pa = 00 we must have cos (a<r) = 0.