BODY BY THE AID OE THE THEOEY OF SCREWS. 
17 
[art. 16] this analogy was generalized into a theorem, which asserted that twists and 
wrenches are compounded by the same laws. We can now show that this theorem is a 
consequence of the reciprocal character of the virtual coefficient. 
3. Source of the identity of the rules for the composition of twists and the composition 
of wrenches. — Let L, M, X be three screws, about which wrenches X, Y, Z equilibrate. 
Take any known screw S m ; let A m , B m , C m be the virtual coefficients of S m with L, M, X. 
If a body receive a small twist u about S m the quantity of energy expended must he 
zero, since the three acting wrenches equilibrate; hut the energy consumed is the 
algebraical sum of <yXA m , aJYB m , soZC m , whence we deduce 
XA m + YB m _{_ zC m = 0 ; 
six of the equations obtained by giving different values to m will determine the values 
of X, Y, Z, and also the conditions which must be fulfilled by the positions and 
pitches of the three screws L, M, N. These conditions we know [art. 16] are only 
fulfilled when L, M, N lie upon the same cylindroid. When six equations of this type 
are satisfied, then all similar equations must also be satisfied. 
Let it now be proposed to examine the conditions under which three small twists 
a, (3, 7 about the same screws L, M, X can neutralize each other ; in other words, that 
the last twist shall restore the body to the same position which it occupied before the 
first. The total quantity of energy expended in the three twists against a wrench F on 
any screw S m must be zero ; hence the algebraical sum of the three quantities, aFA m , 
0FB m , 7 FC m , must be zero, or 
a A m -f- (3 B m -j- 7 C ot = 0 . 
We thus see that the quantities a, 0, 7 must be proportional to X, Y, Z, and that the 
conditions to be fulfilled by L, M, X are precisely the same in both cases. 
4. On the component wrenches about six screws of reference into which any wrench may 
be resolved. — The properties of the virtual coefficient enable us to calculate with faci- 
lity the magnitudes of seven wrenches about seven screws which equilibrate, or, in fact, 
to resolve a wrench into six wrenches about six given screws* [art. 46]. 
Let A x See., A 7 be the given screws, and X, See., X 7 the required wrenches. Let S 
be any screw and B m be the virtual coefficient of S and A m . If the body receive a twist 
a about S, the quantity of energy expended must be zero, and therefore 
BjXj-F &c. +B 7 X 7 =0 ; 
six of these equations would determine the ratios of X, &c., X 7 . By judicious selection 
of S the process is greatly simplified. If S be reciprocal to X 3 &c., X 7 (art. 5), then 
R 3 See., E 7 vanish, and the equation 
BjXj T ft 2 X 2 
determines the ratio of X, and X 2 . 
* The problem of resolving a force along six given lines, where the lines form the edges of a tetrahedron, has 
been solved by Mobitjs (Ckelle’s Journal, t. xviii. p. 207). 
MDCCCLXXIV. d 
