BODY BY THE AID OF THE THEORY OF SCREWS. 
23 
screw reciprocal to 7c screws of the system will be reciprocal to the 7c-\-l screws [art. 36], 
and therefore a screw can be chosen reciprocal to the six screws from which the sexiant 
is formed. 
14. Use of the sexiant in resolving a wrench into components about a group of screws 
with which the wrench is coordinate . — Given 7c screws of a coordinate system of freedom 
of the degree 7c— 1 [art. 31], determine expressions for the wrenches (or twists) « x &c., 
a k about the 7c screws which will neutralize each other. 
Let the given screws be A, &c., A k . 
Take 7 —7c screws X x &c., X 7 _ fc from the group reciprocal to the given coordinate 
system [art. 37]. 
If wrenches a l &c., a 7 about the seven screws A 1 &c., A 7c , X x &c., X 7 _ k equilibrate, 
we must have (art. 12) 
w i o. n 1 
S(A 2 &c., A*, X x &c., X 7 _ fc ) ~ KC - ~ S(A X &c., AnA, &c., X 7 _*) ~ S(A X &c., A*, X 2 &c., X 7 _ fc ) ~ 
where the symbol S denotes the sexiant of the six screws inside the brackets. 
The sexiants under co k+1 &c., a 7 vanish identically (art. 13); hence u k+1 &c., u 7 are each 
zero. The other equations determine the required quantities a x &c., u k . 
15. On a function analogous to the sine of the angle between two vectors . — We have 
pointed out (art. 9) in what respects the virtual coefficient of a pair of screws may be 
considered analogous to the cosine of the angle between a pair of vectors. We hope 
the following attempt to point out a function in the theory of screws in some respects 
analogous to the sine of the angle between a pair of vectors will not be considered to 
transcend the reasonable use of mathematical metaphor. 
The determinant whose evanescence expresses that three vectors are coplanar has 
the sexiant for its analogue in the theory of screws. If the condition that three vectors, 
a, fi, y, be coplanar is satisfied for every vector y, then the sine of the angle between a 
and |3 must vanish. It is remarkable that the vanishing of the sine really involves two 
conditions, for it can only occur when the direction cosines of a and 0 are identical. If 
now the sexiant of A, &c., A 6 vanish for every screw A 6 , the remaining five screws must 
fulfil the two conditions known to be necessary, in order that they may constitute 
members of a coordinate system with four degrees of freedom. If, then, we can find one 
function, the evanescence of which will afford the two necessary conditions, such a 
function may be considered analogous to the sine of the angle between two vectors. 
It can hardly be objected to this analogy that, while two vectors are concerned in 
the one case, five screws enter into the other; for it must be remembered that 2-f-l is 
the complete number of vectors of reference, while 5 + 1 is the complete number of 
screws of reference. 
The investigation of art. (11) indicates the function of which we are in search. If 
the fiye screws are really coordinate members of a lower degree of freedom, the value 
of § must become indeterminate, and therefore 
P 2 -fQ 2 +R 2 =0. 
