SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
47 
9. But proceeding to Q through 
( 1 ) 
clQ 1 dM r , h do 
~nr = ~ A—T ~Tr — cos 0 j— 
dA. 27rA. db J PO^ 
and utilising the integral 
( 2 ) 
r dA _ A +a cos 0 1 
J : PIP ‘PQ’ 
PIP 
fP sin^ 0 + b‘\ 
PR the perpendicular from P on QQ' parallel to AB in fig. 1, 
( 3 ) 
Q^lj(PR)= 
h do 
W’ 
a new form of the III. E.I., not recognisable in the previous expression in (6), § 8. 
We have to make use of the theorems given in the ‘ Trans. American Math. 
Society,’ 1907 (A.M.S.), coimecting the various forms of dissection of ii in the III. 
E.I., and here the relation connecting the incomplete integrals in 0 of ii (MQ) and 
Q (PR) is 
(4) Q (MQ) +1} (PR) = angle between MQP, M(()R 
- _iQN PQ iMN PM p iA4-«cos(I h 
= PR - MQ = PR ■ MQ = asinO 'PQ’ 
as is soon verified by the differentiation ; and for the complete integrals l)etween 0 
and 27r the sum is 27r. 
In il (PR) the dissection of the circle AB would be in strips QQ’ parallel to AB. 
10. Another form for is obtained from the theorem tliat 
(1) (I> + <:2 = 27r 
connects 12, the area of the spherical curve of the cone on the base AB, and 4>, the 
perimeter of the curve of the reciprocal cone, both on the unit sphere with centre at 
the vertex P. 
The section SS’ of the reciprocal cone made by the plane AB is the polar reciprocal 
of the circle AB with respect to the pole M ; a conic with focus at M, and 
(2) SM . MY = ZM. MQ = lA 
The projection of the elementary sector PSS' of the reciprocal cone on the plane 
AB is 
(3) iPS^. . cos PZM = iMS^ dO, 
. . dd> _ MS^ PQ MP^ ^ P(,) -b^l QY^^ h 
^ do PS^ ■ PM "" PY^ PY^ \ FYV BQ' 
