408 The Incomplete Moments of a Normal Solid 
then : f{x,x., . . . w„) =f{y, + r,,h^, y.^ + rji^ , . . . y,^ + n^^i) 
••Vn) 
d 
33/2 
a 
+ .... 
But, (r„l- + r,,-%...)/(y„y,...^„) 
and 
.+ ••• 
= 2 A„ + A,., + . . . ) ;/i + 2 (r„ A,^ + ri,, A^^ + . . . ) ^/^ 
+ ... 
= 2A//, 
dy, "dy, ■ 7 V "3/A ^^3^/= 
and the remaining terms vanish. 
Hence 
fiyi + rJh, 2/2 + rjt, + ...) =f(y„ y,... y^) + 2AA;/, + AV (36). 
Put 2/1 = 0 and 
/(/ii, iCo- *"3 ... a;„)=/(0, iiJa - n'v^h, ■^13^1 + ...) + AV 
say = / {xi, xi ... Xrl) + b^K ( 37 ) 
Hence we may write 
A^^(/(irf-,^3...a-„) = ^ e 2A (38) 
(27r)2VA 
Omitting the first variate from the 2 summation, 
g-7t,V2 AT SA,,-<g + 2Aaia;/x / 
= ^=. = .e 2A 
(27r) 2 ^/A 
Hence I ... I (^2'' • • • ^ dx-^dx^ . . . dx^ 
J— 00.'— ooJ-00 
= 1 I ... (f)i(f)i^~^(l)J'...^J"zdxidxo...dxn 
J — CO J — cc J— GO 
= - j I ... 02'' . . . ^ + ^ + ...j zdxidx^ ... dxn 
. . I <^i''~' 4>2^ ... z{Xi, X2... hs ... Xn) dx-. . . . \dxg'\ . . . dx„ 
(39), 
in which the expressions in brackets disappear. 
\/27 
X an integral of order n — 1. 
