APPENDIX I. 



NOTE I. 



Another solution of the problem of 28. 



LET the intensities of the wrenches on a, ft, ... rj be as usual denoted by 

 a&quot;, ft&quot;, ... 17&quot; respectively. 



As the wrenches are to equilibrate we must have ( 12) 



where A 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&quot; ft&quot; and the other ratios are known. 



But the process will be much simplified by judicious choice of A. If, for 

 instance, we take as A the screw if/ which is reciprocal to the five screws y, 8, e, , rj 

 then we have 



tff y ^ = 0, t3- 5l ^ = 0, HT^ = 0, -ET^ = 0, Sr r)v j, = 0, 



and we obtain 



a&quot;w al j, + ft&quot; -or w = 0. 



Let p be a screw on the cylindroid defined by a and ft. Then wrenches on 

 a, ft, p will equilibrate ( 14) provided their intensities are proportional re 

 spectively to 



sin (ftp), sin (pa), sin (a/3). 

 It follows that for any screw p. we must have 



sin (ftp) CT aM + sin (pa) OT/Jjll + sin (aft) cr p/ot = 0. 



This is indeed a general relation connecting the virtual coefficients of three 

 screws on a cylindroid with any other screw. 



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, ft) which is reciprocal to if/. 



Then 



Tp* = 0, 



and we have 



sin (ftp) iy a&amp;lt;il + sin (pa) CT^ = 0. 



31-2 



