BODY BY THE AID OF THE THEORY OF SCREWS. 
25 
Before commencing the proof we remark that, 
If an impulsive wrench F acting on a rigid body about the screw X be capable of 
producing the unit of twist velocity about A, then a wrench 1A> about X will produce 
a twist velocity a about A. 
Let X„ X 2 , X 3 , X 4 be four screws on the impulsive cylindroid, the wrenches appro- 
priate to which are F,^,, F 2 <y 2 , F 3 <a 3 , F 4 &/ 4 . Let the four corresponding instantaneous 
screws be A„ A 2 , A 3 , A 4 , and the twist velocities are &> 2 , at 3 , <y 4 . Let <p m be the angle 
on the impulsive cylindroid [art. 7] defining X m , and let 0 m be the angle on the instan- 
taneous cylindroid defining A m . 
If three impulsive wrenches equilibrate, the corresponding twist velocities neutralize : 
hence it must be possible for certain values of u x , co 2 , <y 3 , to satisfy the following 
equations [art. 10 ] : — 
1 
sin (0 2 — 0 3 ) — sin (d 3 — 
“sin ( 0 , — 0 2 )’ 
®>1 F 2 W 2 
F 3 CU 3 
sin (9 2 -9 3 ) sin (93-9,) 
~sin ( 9 ! -9 2 )’ 
C0 2 °°3 
w 4 
sin (0 3 — 0 4 ) sin (0 4 — 0 2 ) 
sin (0 2 — 0 3 )’ 
F 2 w 2 F 3 «j 3 
F 4 W 4 
sin (^3-94) sin(9 4 -9 2 ) sin (<p 3 -<p 3 )’ 
whence 
sin (0,— 0 2 ) sin (0 3 — 0 4 ) sin (9,— <p 2 ) sin (<p 3 —<p 4 ) 
sin (6 3 —fl,) sin (0 4 — 0 2 ) sin (9 3 — 9i) sin (<p 4 — <p 2 ) ’ 
which proves the theorem. 
If, therefore, we are given three screws on the impulsive cylindroid and the corre- 
sponding three screws on the instantaneous cylindroid, the connexion between every 
other corresponding pair is geometrically determined. 
19. On the impulsive wrenches which arise from the reaction of constraints . — Whatever 
the constraints may be, their reaction produces an impulsive wrench It, upon the body 
at the moment when the impulsive wrench X, acts. The two wrenches X, and It, com- 
pound into a third wrench Y,. If the body were free, Y, is the impulsive wrench to 
which the instantaneous screw A, would correspond. Since X,, X 2 , X 3 are cocylin- 
droidal, A„ A 2 , A 3 are cocylindroidal (art. 17), and therefore also are Y„ Y 2 , Y 3 . The 
nine wrenches X„ X 2 , X 3 , R„ R 2 , It 3 , — Y„ — Y 2 , — Y 3 must equilibrate; but if X„ X 2 , 
X 3 equilibrate, then the twist velocities about A,, A 2 , A 3 must neutralize, and therefore 
the wrenches about Y„ Y 2 , Y 3 must equilibrate. Hence R„ R 2 , R 3 equilibrate, and are 
therefore cocylindroidal. 
Following the same line of proof used in art. (18), we can show that 
If impulsive wrenches act about any four cocylindroidal screws upon a partially free 
rigid body, the four corresponding initial reactions of the constraints also constitute 
wrenches about four cocylindroidal screws ; and, further, the anharmonic ratios of the two 
groups of four screws are equal. 
MDCCCLXXIV. 
E 
