380 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
The corresponding theorem in two dimensions, which we will also employ, is this :— 
The integral 
taken round the periphery of a circle of reference, is 
IP 
Hence it follows that 
[v*, 
taken round the same periphery, is 
„ * R 2i id? , d 2 \\ r 
,rB 72 H\ iifh? r dy) V ° 
W 
( 5 ) 
The theorems expressed by (3) and (5) are employed below in a variety of cases con¬ 
nected with spherical harmonics. It is therefore necessary to specify at the outset 
the notation to be used, 
We denote, according to custom, the colatitude and longitude of a point referred to 
the axes of reference by 9 and <f>: cos 6 and sin 6 by p, and v respectively. We denote 
the Legendre’s coefficient or zonal harmonic of the i a degree by the symbol P, so 
that Pi is defined by the equation 
«!P,= (-i)V«| l .(6) 
According to Clerk Maxwell’s theory of poles the general expression for the har¬ 
monic of the i th degree is given by the equation 
H Y;= (-!)'» 
w+ 1 _ 
di 
d/qd/q ... dlii r 
(?) 
where — means differentiation with regard to an axis h. 
dh ° 
In the harmonic (7) the i poles are the points where the i axes of differentiation cut 
the sphere of reference. In the harmonic (6) the i poles all coincide with the point 
where the axis of z cuts the sphere. If the % poles coincide at any other point the 
harmonic will be denoted by Q;. 
In the tesseral and sectorial system we have ventured to depart from the usual 
notations by denoting them as follows :— 
