APPENDIX I .
NOTE I .
Another solution of the problem of § 28.
Let the intensities of the wrenches on a, ß, ...t) be as usual denoted by
a", ß", ... y" respectively.
As the wrenches are to equilibrate we must have (§ 12)
a ß"^Bl + f'^yX + 8"^SÅ + «"WeX + + f'^X = °,
where Å is any screw whatever.
If six different but independent screws be chosen in succession for A we have
six independent linear equations, and thus a" - ß" and the other ratios are known.
But the process will be much simplified by judicious choice of Å. If, for
instance, we take as Å the screw i// which is reciprocal to the five screws y, 8, e, £,
then we have
w ytp — 0) eifi —‘ 0, w — 0, — 0;
and we obtain
a'vr^ + ß"^ß^ = 0.
Let p be a screw on the cylindroid defined by a and ß. Then wrenches on
a, ß, p will equilibrate (§ 14) provided their intensities are proportional re-
spectively to
sin (ßp), sin (pa), sin (aß).
It follows that for any screw p. we must have
sin (ßp) ■sra/i + sin (pa) + sin (aß) = 0.
This is indeed a general relation connecting the virtual coefficients of three
screws on a cylindroid with any other sci-ew.
Let us now suppose that p. was the screw just considered, and let us further
take p to be that one screw on the cylindroid (a, ß) which is reciprocal to ip.
Then
= 0,
and we have
sin (ßp) + sin (pa) -tsß^ = 0.
31-2