404 The Inmmi^lete Moments of a Normal Solid 
3. Gentroid of a normal quadrant found hy geometry. 
In this and the following section I shall show how the lower moments may be 
found by direct methods and also how these methods lead to results of such com- 
plexity with high moments that a symbolic method is necessary. 
\ 
B \C 
I , 
'\ 
"\ 
\ 
T 1 
1 1 . 
\y, i 
1 1 
V ^ X 
t 0 
\k 
E 
\ 1 
Let OX, OY be the co-ordinate axes through the mean of a normal solid and 
let AD, DE be the projections of the bounding planes of a quadrant. AB is the 
projection of a plane cutting the solid perpendicularly to the Y axis and at a dis- 
tance y — y^. The equation of the ciu've in which it cuts the surface will be 
Nz = N'-j^.- ,^ .e 2Wr-.^y (13), 
V 27r vztt V 1 — r- 
i.e. a curve whose mean is at x = ry-^, whose standard deviation is Vl — r^ and whose 
area is NK. Considering B to be at an infinite distance the area of BA is 
nI z {x, y^) dx = a say, (14), 
and since the centroid of a segment of a normal curve 
ordinate . , i i v> /i -\ 
= X (stand, dev.)- (I-)), 
area 
writing the centroid of BA as we have 
^,,-ry, = -fc^^'^^') (16). 
Therefore the x moment of the section AB may be written 
Xy^a = ry,a- (I - r-) . z{li, y,) (17). 
