Ball — Note on a Determinant in the Theory of Screivs. 377 
six forces to equilibrate. Hence it follows that the determinants 
analogous to that just written, but with five and six rows of elements 
respectively, are all zero. 
These theorems simplify our expansion of the original harmonic 
determinant. In fact, it is plain that the coefficients of x"^ of x, and 
of the absolute term vanish identically. The terms which remain are 
as follows : — 
x^ + Ax^ + Bx^ + (7^3, 
where A = '^—. 
Pi 
cos (13) 
cos (23) 
cos (13), cos (23), 1 
If by S (123) we denote the scalar of the product of three unit 
vectors along 1, 2, 3, then it is easy to show that 
^2(123)= ^,23. 
We thus obtain the following three relations between the pitches 
and the angular directions of the six screws of a coreciprocal system, 
namely, 
Pip2 
3) _^ 
PiPzPs 
The first of these was given in the original "Theory of Screws," the 
second and third I now print for the first time. 
In the " Theory of Screws," p. 148, I remarked : " We are thus 
presented with no fewer than six formulse involving the pitches and 
E.I.A. PEOC, SEE. III., VOL. I. 2 E 
in which 
-S'i23 — 
p^plpi 
1, cos (12), 
cos (12), 1, 
