292 
AiE. E. W. BAENES ON THE THEOEY OF THE 
Then it is readily seen that 
("i + "2) — = log (w| + "0) — log 6J| — 2 miTL 
logy^ (wj + ct)^) — logy^oj^ = log (wj + w;) — log Wo — 2 m ttl 
logo,., (wj + CO. 2 ) — log^^Wo = log (co^ + Wo) — log Wo — 2m'TTL 
l^&%("i + ~ l*^gY"i = log(w| + Wo) — log wi — 2mTn. 
By inspection of a diagram we see that m and in i^otli vanish if the difference ot 
tlie amplitudes of w^ and Wo is less than tt, these amplitudes l)eing measured between 
0 and hb positively or negatively from the positive lialf of the real axis. In 
jjarticidar when the real parts of Wj and Wo are both positive, m and m' both vanish. 
Not only so, but in all cases either m or m' must vanish. 
Again, if the differeiice of the amplitudes of Wj^ and Wo is greater than tt, m and m 
cannot l)oth vanish. In fact, in this case we liave the important relation 
in 
m' = ± 1, 
! Wo 
the upper or lower sign ijeing taken according as Is negative or positive. This 
result is intuitive geometrically ; in tlie figure, for instance, two cases are indicated 
in which I 
Wo 
Wi 
IS negative. 
For corresponding to the unaccented value of w.,, 
in = L 
m' = 0 
and corresponding to the accented value of Wo, 
in = 0 
m' — — 1 
Tlius in both cases in — m' — 1. 
No such simple expression can be given for rn + in, a numlier wliich is of constant 
occurrence in the liigher theory. 
However, from the values for m and in previously given, we see that when the 
axes of Wj and Wo include the axis of —1 within an angle less than two right angles, 
the values of [m + in') are given liy the table 
