189 
Let Z be the projection of the line L on the axis of 0 ; also 
suppose that there are n sections. Then equation (1) may- 
be written 
A*i + Az2 + .. • • A zn Z _ (A x + A 2 4 - . . . Aw) L 
. — ooby r- — 
n L n 
or more briefly 
4sA a AZ = ^2AAL (4) 
Z Ju 
Z L 
when AZ and AL have been written for - and - respectively. 
n n 1 J 
Operating in a similar manner on equations (2) and (3) we 
shall have 
y2A a AY = — ^SAAL (5) 
isA I AX = ~2AAL (6) 
Now suppose n to become indefinitely large, then the limits 
of the expressions (4), (5), (6) are 
\ j\dZ^ J\dh (7) 
o o 
i f\dY = C ^ f\dh (8) 
o o 
i J X AxdX = ^ j\dL (9) 
o o 
Now suppose a solid to be constructed whose axis is perpen- 
dicular to the plane of xy and of length Z and whose 
sections parallel to the same plane are A zl , A. z2 , &c. Let 
the volume of this solid be Y z , Let similar solids be con- 
structed whose axes are Y and X. Let these be denoted by 
Y y , Y x , then we shall have 
( 11 ) 
