148 Mr. Boots on a certain Multiple Definite Integral. 
Now — fan 
Ss ce: BR 
do a ie eet Gee 
; Sees au d é : , 
Substituting this in the place of ds et) in (24), changing s into A, and re- 
jecting the integral sign, because all the other elements vanish, we have 
aves 5h, h h,na 
* da r(A+MA)[(A-M!) bh) +h) a | (26) 
which is a finite expression for the attraction of a pu oe mets ellipsoid on an 
external point abc in the direction of a, the law of force bene 7 
If h, =h, =h,=r, and if we make a’+-6?+-c’ = a, we find A = d* — 7", 
and substituting and reducing 
dv Anar® 
—i__ = —_- — 2) 
°da — 30(P—r) (27) 
and this is the result to which we should be led by direct integration, the ellipsoid 
becoming a sphere. 
If a, b,c, or any of them, be very great, we find A = d’, and substituting and 
reducing 
dv Anh hah, a Aa 
it —— — 7 
° da 3 ad? Ape ee) 
where a is the solidity of the ellipsoid. This shews that at a very great distance 
the attraction is the same as if the mass of the ellipsoid were collected at its centre. 
If abc be a point in the surface of the ellipsoid, A = 0, and the expression 
for the attraction is infinite. 
Let us, in the next place, suppose the density variable, the attracted point 
being still external. 
Reasoning as in the last case, we ue that when s =A, the numerator 
d ds 3h Os ei 
ds = — nk (c) in (24) assumes a finite value - == adi 1). This will give rise to a sepa- 
mee ite term. From s=A tos=o ws quantity under the sign of integra- 
River : ; d s 
tion is continuous. Putting, then, det (*) =f (a); we get 
o 
