Mr. Boor on a certain Multiple Definite Integral. 145 
But 
digde dh eae, 
da” dads” st+hjdo’ 
hence, 
dy a2 Madsfi(s). 1s 
P? a = 2M fs A (sh?) [(s--h2) (sEh2) (s--heyy OO) 
To the limits c= 0, « = 1, correspond the limits s = #, s =A, A being given 
by the equation 
a b c 
==". 18 
A+h? + Ath, 0 Ath; oe) 
Now the integration in the second member of (17) extends to all positive 
values of s which make o fall see 0 Be “e i e. to all positive values of s 
which lie between A and ow. Let 7 a s+ ie be greater than 1, which im- 
h’ 
plies that the attracted point is external ; o is then positive, and the limits of s 
are accordingly A and #; whence 
Fs tari) oa ey GEIST (19) 
; (is baat i . : : 
Again, let mE + aot = be equal to or less than 1; A is then 0 or negative, 
1 2 Us 
and the limits of s are 0 and »; whence 
ds f (c) 
(EAE) [(SFhe) (8-FAZ) (8 FRE 
This formula determines the attraction in the direction of the axis # when the 
(20) 
dv 
— = dah hh 
attracted point is on the surface or is internal. 
‘To deduce from (19) the common expression for the case of external attrac- 
tion, we must assume 
h; a ei ant 
h? —- s 
and transforming, we get 
= 4h hhna u-duf (c) (21) 
= Atha Waeee =f (hig hey) (MF hep (Ae — hse) 
in a 
VOL. XXI. U 
