410 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
As a limiting case, let the ellipsoid wear down to the focal ellipse, and for the sake of 
brevity let the semi-axes of that ellipse be denoted by y’and g. We find 
(44) 
These results may be further simplified if the ellipse becomes a circle. 
The Potential clue to an Ellipse of uniform density. 
§ 24. The integral 
j'e <u " H typds 
taken round the perimeter of an ellipse of semi-axes a, b, is derivable from the corres¬ 
ponding case of the circle of radius It by the theory of corresponding points, and the 
result is 
thi w 
if we write herein cid, bO, and recollect that y>Sd= dSp>, then 
j j e^+fydxdy 
taken over the area of the ellipse is equal to 
jje^+^l'dsdd 
the limits of 9 being from 0 to 1. The result of integrating this is 
(45) 
2 t TCib$ 
(«V + J 2 /3 2 ) ; 
o 2 2i+1 i ! i + 1! 
Similar reasoning would give us 
o o 
+' 
r>a.i'+ 
dxdy 
(46) 
Confining our attention, however, to the result just found (46), we see that the 
integration over the area of the ellipse of any function V of x, y, having finite differen¬ 
tial coefficients at the origin, leads to 
