273] EMANANTS AND PITCH INVARIANTS.
287
If we substitute these values for alt ... an in the expression
pa=tp1a^
we obtain the equation
0=p.2
cos2 al cos2 a2
------
. Pi P*
cos2 a6
P« .
+ 2po
cos al ( pT cos al — dal sin al)
Pi
cos a6 (pe cos a6 — daS sin a6)
+
P«
- 2
+
( p1 cos a 1 — dal sin al)2 ( p6cos a&-dals sin a6)2
Pi P«
As al, &c., daj, &c., p{, &c. are independent of pa we must have the three
co-efficients of this quadratic in pa severally equal to zero.
272. Three Pitches Positive and Three Negative.
The equation
cos2 al cos2 a2 cos2 a6
------■+-------+••. + •-----=0
Pl Pl Ps
also shows that three pitches of a set of six co-reciprocals must be positive
and three must be negative. For, suppose that the pitches of four of the
co-reciprocals had the same sign, and let a be a screw perpendicular to the
two remaining co-reciprocals, then the identity just written would reduce to
the sum of four positive terms equal to zero.
From this formula and also
1 1
—I----
Pi Pi
1
P