418 The Incomplete Moments of a Normal Solid 
These when expanded by means of the tables will give a series of" terms such as 
rk rh 
(o) Uif'z (h, y) dy and (b) x^'t"'z (x, k) dx. 
J - 00 J —CX) 
rk 
(a) may be regarded as the {lm)th. moment of z{h, y) dy, tha,t is of HA (see 
J — GO 
equation 7), and (6) may be regarded as the (r?u')th moment of I z{x,k)dx,th3it 
J — ad 
is of KB. That is they are moments of the bounding surfaces of the quadrant 
(equation 8). 
If we write the moment coefficients of HA and KB as Ai,n and Bi',n' respectively 
we may write 
f Uf-z (h, y) dy = HA. At,, = HA . IM,,, (86), 
J —00 
r xik''''z{x,k)dx = KB.Bi;„, = KB.t"-'Bi', (87). 
It remains therefore to find a means of evaluating A^m and B^q. 
An alternative form of the expansion will be got by expanding a yfr instead of 
a (f) but the ultimate results will be identical. 
To take an actual example 
rk rh 
I x^yzdxdy 
k rh 
(^-yjr + yjr 4- 2r<^) zdxdy 
dx 
dx 
^ ~ 11 /-oo 1^'"^' + r + 2r)^ + (c/>= + 1 + 2r=) ^| zdxdy 
rk rh 
{r(f>- + 3r) 2 {h, y) dy - (0- + 1 + 2r^) z {x, k) dx 
J - 00 J — 00 
= - r irle - r + 3?-) z (h, y) dy - f (x- - 1+ 1 + 27-'-) z (x, k) 
J — CO J —00 
rk rk rh rh 
= - r h-z(h, y)dy- 2r z {h, y)dy - x"z {x, k) dx - 2?'^ I z {x, k) 
J - 00 J — CO J —CO J —CO 
= - rh'HA . - 2rH .A-KB.B^- 2r'KB 
= -HA (r¥ + 2r) - KB {B,, + 2r-) (88). 
For theoretical purposes the development may be treated more systematically 
as follows. 
Since [ h'y'^zih, y) dy^HA. Ai^ (89), 
J ~ CO 
rk ■ rk 
it is clear that </'^~^9 ^ V) = T^^^^ q{h, y)z {h, y) dx 
7—00 J — 00 
= HA.T,^,,^A (90), 
in which the T will be expanded in A's having the same suffixes as the /t's and ?/'s. 
