312 
Mil. E. W. BAKXES OX THE THEOEY OF THE 
But we have seen in § 22 that, if the difference of amplitudes of oj^ and w., be 
greater than tt, 
r/i — B 
tlie ii})per or lower sign being taken according as l( — | is positive or negative. 
Similarly it may be proved that if the difference of amplitude of ojj and — cu.. 
IS > TT, 
1 ) 1 ^ —9/i/ = rh 1, according as l(^j i« +''r or —re. 
And .— and is > tt. 
— Tlln — L . . . . 1 ( ~ I IS ff- V6 or — V6 
\"l/ 
while, ffiially, .~ 
7/«3 — = d- 1 • • • • l j is ff-re or — rr. 
In all other cases the differences between cori'espondiiig ms vanish. 
But the difference of amplitude of one, and of only one, pair of the sec 
fajj, oj.^ ; oj^, — w.,; — ojj, (X)., i — oj 
V 
■CO., 
can be greater than tt. 
Hence — (m — m) + (/Uj — in^) -f {m.^ — m.^) — {rn.^ — m..') 
must always ecjual i 1. 
And it is easy to see, by taking tlie particular cases which can arise, that the 
upper or lower sign must be taken according as is positive oi' negative. 
We have then ffnallv 
V- 
0), 
-'^ = ± 
'.TTL 
and tlierefoi’e 
■>7107^ — ~ i 
the upper or lower sign being taken according as l(^) B l)ositive or negative. 
1/ 
^ 35. For brevity, we merely indicate the relation of the formuke which Ave liave 
found to the known relations : 
(T {z OJo) — — C 
(-f) 
cr(:) 
cr(2). 
By §3 
c" 4' Wj) _ —ti-uji — v ——L ~ _j_ 
< t { s ) ' : 
"I 
1 
'^{z + Wil +fWp + COo)] 
L lb (-1 + J 
11 ^ A 
+ Wj) n B 1 i «:) 
Ti ± wj) 
Fi ’ I i ^ 
