BODY BY THE AID OE ^HE THEORY OE SCREWS. 
21 
III. THE SEXIANT. 
11. Definition of the Sexiant . — When six screws, A 1 &c., A 6 , are reciprocal to a single 
screw T, a certain relation must subsist between the pitches and the coefficients of 
A! &c., Ag. The function which, when equated to zero, gives the condition required, 
may be called the sexiant of the six screws. 
The equations of the screw A* are 
x—xjc y—yi c z—zk , i x ^ 
— =^r='vr(P ltch= ^- 
The reciprocal screw T, which without loss of generality we may suppose to pass 
through the origin, is represented by 
;=H (Pitchy). 
The condition that A* and T be reciprocal is 
(g + §k)( aa k + (3(3 k + yy*) + x k { y(3 k — (3 y k ) -\-y k {ay k — yu k ) -f z k ((3a k — a/3*) — 0 . 
Writing the six equations of this type, found by giving Jc the values 1 to 6, and eli- 
minating the six quantities 
§«> S(3, ft 7> 
we obtain the result : — 
+ ft*^ 
A?i+¥-yA, 
yifi+ftffi— «#!, 
a j, 
ft, 
7i 
a 2?2 - l - y2^2 ft^25 
Pa§a + C&aZ a — </ 2 X 2 , 
72?2+fcx 2 —u 2 y 2 , 
«2, 
ft. 
72 
^z+7zy 3 (3 3 z 3 , 
(3 3 ^3~{~ a 3^3 y 3 ^> 
7af 3 (3 3 x 3 a 3 y 3 . 
(3 S , 
73 
a 4g4+ 7$* — ft 2 45 
(3^i+06 4 z 4 — <y 4 x 4 , 
7&+P4X4—UM, 
Ci 4 , 
ft, 
7k 
«5§ 5 +y 5 y5— 
P s §5 + <*ePB — 7ipH 
7*8 5+(3^5—a 5 y 5 , 
*5, 
(3 5 , 
7s 
®6§6 “t" 7ol/6 @6^6: 
ft§6 + a 6 2 6 — 76^6) 
7e§6+(3 6 a; 6 —a 6 y 6 , 
<*6, 
ft, 
76 
By transformation to any parallel axes the value of this determinant is unaltered. 
The evanescence of the determinant is therefore a necessary condition whenever the six 
screws are reciprocal to a single screw. 
The sexiant is only a function of the differences fi — f 2 &c. of the pitches; and when 
the pitches are equal the evanescence of the sexiant expresses that the six lines which 
form the screws are in involution. 
The property possessed by six screws when their sexiant vanishes may be enunciated 
in different ways, which are precisely equivalent. 
(a) The six screws are all reciprocal to one screw. 
(d) The six screws are members of a coordinate system of five degrees of freedom. 
(c) Properly selected wrenches about the six screws equilibrate, when applied to a free 
rigid body. 
(d) Properly selected twist velocities about the six screws neutralize, when applied to 
a rigid body. 
