A. Ritchie- Scott 
405 
Now let the surface BA move from y = — x to y — then Xy^a will generate 
the X moment of the quadrant, y^a will generate the y moment and z(h,y^) will 
generate the area of the surface exposed by the plane AD. Writing the centroids 
of the quadrant x,y 
af = 7-«^-(l - r^)f' z(h,y)dy (18). 
.(19). 
A similar discussion of the y moment will give the equation 
ay = rax — (1 — r") I z (x, k) dx 
Both the integrals are known and the centroids may be found as the solution of a 
pair of simultaneous equations. 
4. Another method of finding the moments. 
The method of § 2 may be expressed analytically and extended to higher moments. 
For example second moments may be found as follows : 
d'^z 1 
d^z 1 
Hence 
Sy' {l-r-J 
^ ^r^x' - 2rxy + f - (1 - r'^j z {x, y) 
,..(20). 
{x'^ — 2rxy + r-y'^) z {x, y)— (1 — r'^)- 
dx' 
d^z 
(1 - r') z {x, y) 
{rx' - (1 + r^)xy + ry-\ z{x, y)= - (1 - + (1 - ^ (-'f, */) 
(j2rfS. _ 2rxy + 2/^) z {x, y) = (1 - r')- ^- + (1 - r') 2^ {x, y) 
Multiplying both sides by N and integrating, and writing 
m .m2g, = N ^^x' . z {x, y) dxdy^ 
m . mi, = N I jxy . z (x, y) dxdy | 
etc. 
_ _ _ iV" f[d^2 \ 
we get mao — 2rm„ + ?-^TOo2 = ^ ^"'^ '"'^ / j dxdy + {I — r^) 
N 
.(21). 
•(22), 
rma, - (1 + r^) m„ + rTO„2 = " " | | ^'''^3/ + r (1 - r^) > . . .(23). 
- 2rmji + = + ^ ( 1 - r'-)- dxdy + ( 1 - / -) 
Biometrika xiii 
26 
