as are terminated by Planes, &c. 287 
Hence the attraction of a prism, whose height is d, and base 
a regular polygon of n sides, composed of triangles having 
such a law of density as was supposed above, will be, on a 
point placed perpendicularly over its centre, 
» . /■» arf ( a,T ) T f 
A _ 2 naj ^ Z + T%) (a * +(1 + r *)T >) t 
This expression would be easily integrable on various supposi- 
tions. Thus we might conceive the density at the ordinate ks 
to vary as the line ps, drawn from the attracted point to its ex- 
tremity s ; this would be to make f ( a,T ) — V al-\- (i + O T'; 
whence A = vna J * p—p = rnaa [ L . (a 2 -f- T 1 ) — L . a* j . 
Again, we might suppose f (a,T) = a 2 T 2 = ( pk ) 2 ; this 
would give A = indf- -^|S== = ^ { v/a‘+(i+,-yr- a } . 
But the kind of problems w r e are engaged about does not re- 
quire us to know the value of A itself ; its fluxional coefficient 
with respect to T being alone wanted, and this is always 
2narf ( a,T ) T 
x a 
( 0 - 
(a 2 + 'l z ) vV+ (1 + r l ) T z 
For suppose we had actually found the fluent; when we make 
use of it in such a problem as the last, we must change T into 
y, and take the fluxion with respect toy, and the result must 
necessarily be 
2 liar? (a,y) y 
x a ; which we might have 
(aH/) v' «*+ (1+^) y 7, 
arrived at simply by changing T into y in the expression 
marked (1). 
Lemma 2. 
To find the quantity of matter in a right prism, whose base 
is the triangle rmv, fig. 1, and height d ; supposing the den-* 
sity at any ordinate ks to be f ( a,T ), 
P p 2 
