APPENDIX I. 



489 



If all the first minors of the discriminant of S vanish we must have the 

 following identity 



nn n &amp;gt; 



by which we have 



S = A 33 s 3 2 + A^s? + 2A u s 3 s 4 ... 4- A nn s n 2 . 



Hence U + XV = A^s.* + ^ 44 s 4 2 + 2^348^4 . . . + A nn s n 2 . 



In the case of n = 3 we have 



which proves that V and U have double contact as we already proved in a different 

 manner. 



In the general case all the differential coefficients of S will vanish if s 3 =Q...s n ~0, 

 but these latter define a cylindroid and therefore whenever the discriminant of s 

 has two equal roots, every screw on a certain cylindroid is a principal Screw of 

 Inertia. 



If the discriminant had three equal roots then S could be expressed in terms of 

 s 4 , ...s n and in this case every screw on a certain 3-system would be a principal 

 Screw of Inertia. 



If n 1 of the roots of the discriminant were equal, then every (n 2)nd 

 minor would vanish, S would become the perfect square s n 2 to a factor. 



And we have 



In this case every screw of the n - 1 system defined by s n = will be a principal 

 Screw of Inertia. 



NOTE III. 



Twist velocity acquired by an impulsive wrench, 90. 



The problem solved in 90 may be thus stated. 



A body of mass M only free to twist about a. is acted upon by a wrench of 

 intensity tf&quot; on a screw rj. Find the twist velocity acquired. 



From Lagrange s equations we have, 86 



d fdT\ dT 

 - = 



