SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 4? 



9. But proceeding to Q through 



1 dM 



and utilising the integral 

 (2) 



PR the perpendicular from P on QQ' parallel to AB in fig. 1, 



/ O v 'WDTA _ f Aa cos + a 2 cos 2 # &d$ 



V s J~ ~PP7~ 'PQ' 



a new form of the III. E.I., not recognisable in the previous expression in (f>), 8. 



We have to make use of the theorems given in the ' Trans. American Math. 

 Society,' 1907 (A. M.S.), connecting the various forms of dissection of 12 in the III. 

 E.I., and here the relation connecting the incomplete integrals in 6 of 12 (MQ) and 

 Q (PR) is 



(4) Q (MQ) +Q (PR) = angle between MQP, MQR 



.,QN PQ .,MN PM ._! A + acoBfl h 



Sill T-.T-, - -*rt~\ COS T-VT> -, f,^. COt . lw , , 



PR MQ PR MQ a sin PQ 



as is soon verified by the differentiation ; and for the complete integrals between 

 and 2x the sum is 2?r. 



In Q (PR) the dissection of the circle AB would be in strips QQ' parallel to AB. 



10. Another form for 12 is obtained from the theorem that 



(1) * + Q = 2 T 



connects 17, 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 = b 2 . 



The projection of the elementary sector PSS' of the reciprocal cone on the plane 

 ABis 



( 3 ) iPS 2 . d# . cos PZM = MS 2 d6, 



d* _ MS 2 PQ MP 2 _ PQ . b _ (. . Cm A 

 dO == PS^ ' PM == PY* = PY 2 " PYV PQ' 



