.MR. Iv M'. BARNES ON THE THEORY OF THE 
O 
•) 
■_R) 
Avhere M = 0, unless the axes of — and ("i + cj.i) include the axis of — 1, in 
ij 
which case 
M = d- 1) the upper or lower sign being taken as I(ajj + u>n) is po.sitive or 
negative. 
§ 40. Let us take now the integral which has just been defined, and apply to it 
the procedure of § 38. 
We shall evidently have to consider integrals of the type 
«r(l - s) 
27r 
where the axis of n lies within the smaller angle between the axes of and oj.^. We 
can at once see that tins integral 
iFd - s) 
zy-\h 
where a = 0, unless the axes of L and - 
/t 
embrace the axis of — 1, in which case 
g = d; 1, the upper or lower sign being taken as I(u.) is positive or negative. 
Let n = re'\ wo = he'^, where 0, a and /3 are measured betAveen 0 and 
277 by rotation in the positive direction from the positiA’e half of the real axis. The 
axis of L proceeds from the origin to the point + therefore AA’here 2 : is at a 
distance p along tlie axis of L, 
nz — rp e “ 2 “/- 
This quantity has its real })art positiAm Avhen 
a relation Avhich is satisfied Avhen 6 lies betAveen a and /3, and the difference betAveen 
a and /3 is less than 77 . It is also satisfied Avhen the axis of L proceeds from the origin 
to the point AAfiiere e is a quantity less than half the excess of 77 oA^er 
a ^ (B. We see then that the axis of — lies Avithin a range of a right angle on 
either side of the axis of L. Hence by the propositions preAuously proA^ed [“ Theory 
of tlie Gamma Function,” §§ 33, 34], 
H Gl - s) 
-1 
dz = 
iTil-s) 
ilT 
n 
-1 
dz, 
