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,
|" cos («3) = y"yi + p"pt>
l