A. Ritchie-Scott 
419 
Hence we may write immediately 
I* r <l>i'ylrizdwdy = - HA. Tj,^,^gA-rKB.Tp_,^rjB = -Tp^,,g3I ...(91), 
^ — X J - X 
or reducing the yjr 
= - rHA . 7V,,_iI - KB . 1\,_,B = - T,^,^,M', 
where M or M' may be regarded as a kind of complex moment coefficient. 
Applying this to oar example we have 
fit ch 
(</)--<|r + -v/r + 2/'^) zdxdy 
— 00 - 00 
=^-[T,,{M')+l\{M') + 2rT,M] 
= -{rHA.T,o{A) + KB.T,,B] 
- (rHA + KB) - 2r {HA + rKB) (92). 
But T,o (A) - Z„ - 1 = - 1 
and . T,,(B) = B,,-1, 
and the whole reduces as before to 
- HA {rU + 2r) - KB (B,, + 2r'). 
10. Evaluation of the moments of the houndmg surfaces of the quadrant. 
In equation (7) the area of the surface bounding the quadrant a and parallel to 
the y axis was represented by the symbol 
H . „ 
N^A = iyI z'{h',y')dy', 
O'x J -00 
and that parallel to the x axis by 
N~B = n\^ z'{x, k')dx. 
Consider now the pqth moment of say the latter, the B face. It will be repre- 
sented by 
N I x'Pk''J2'(x', k') dx' = JSfk'i f x'Pz'ix, k')dx (93), 
J — 00 J — X 
since y' = k' over the whole face. Representing this as the product of the area and 
its pqih. moment coefficient we may write 
x'Pk''iz'{x',k:)dx'=N — B.B\,,j = N—B.k"i.B', (94), 
J -X CTy O-y 
and it is the value of the coefficient 5'^,o that we have to determine 
^ [x- . z' (x, k')} = nx'-^ . z' {X, k') + X- I - "^J^^'y'l^^y]^ z' {x', k') 
-r^.—^~^z{x\k') ...(95). 
0-^2 (1-r-) o-,, ■ 0-^(1 - r-) 
