Mr. Boots on a certain Multiple Definite Integral. 149 
dy tah, h,h,af( ] ) 
—t—= SARE 5 Sore =  @ 7 Sve.” ay oe 
5 da A(A+h,") [(A+ h,’) (A+ A”) (Ah) }! Ee + (+AS? =F = 
‘ dsf'(a) hed (29) 
S(6E AY) [6+h2) (8 FAY) (sh) 
This expression consists of two parts; the first, which is finite and algebraic, 
snows what the attraction would be if the density throughout were uniform, and 
equal to the density at the surface; the second expresses the effect due to the 
excess of density above, or defect below this uniform state. 
When the particle is internal the finite algebraic term must be rejected, and 
the lower limit of integration in the remaining term must be replaced by 0. 
4 
—trhh,h,a 
POSTSCRIPT. 
The value of the multiple definite integral, 
ze re Lee 
-=\ dade, S (GEA GE +E) 
[(4,—2,)?"+ (@,—2,)?.--- (@,—4,)°" Ew} 
subject to the same limiting conditions as before, will be expressed in the same 
form (12), with this difference only, that 
9 2 9 2 
as a; aa u 
ee ne a Stay aie 
§ 
s 
Apparently this is the most general theorem of the kind; for the expression 
Z(a-+ bx-+ ca’) may, be reduced to the form exhibited in the denominator 
under the index 7. 
Lincotn, Dec. 1845. 
VOL. XX1I x 
