A. Ritchie-Scott 
417 
TABLE II {continued). 
6. (^«(^) =(,^'«- 15^ + 45^^^-15)^ 
<^5^ (z) = {x^y — IQx^y + l^xy — 15r — brx* + ZOrx'-) z 
(f)^f2 (z) = (xy - QxY - 8rx^y + 6(1 + 2r^) x^ + Sy^ +-24<rxy - -3(1 + 4/'-)) z 
(fi»yjr^ (z) = (a^y^- Sx^y- Sxy' - drxy +9(1 + 2r-) xy + 9rx- + 9r/ - lir (3 + 27'^)) z 
7. <^T{z) =(^'-21a.-5 + 105«»-105a,')^ 
^6-^ (0) = (af-y — \hx^y + 45./;-?/ — \^y — ^a^r + GOar'r — %^xr) z 
(f)^f^ (z) -- {xy - lOA'-y + 15a;?/" - lOrx'y + 60/'a;=?/ - 30ry 
-15(1 + 4r-) + 10 ( 1 + 2r-0 xi' - a;-^) 2 
{z) — {x^y^ — Qx'^y'^ + 8?/^ — 12ra^?/- + 36ra;?/- — Sa;-"?/ 
+ 18(1 + 2r^) xi'y -9(1+ 4r=) 7/ - 12?- (3 + 2?-=) a; + 12r«0 ^ 
8. <^8(^) =(a;«-28a;« + 210«^-420a;2+105)2 
(^) = (a;7t/ - 2lafy + lOSa'^?/ - lOoxy - 315ra;- + 105?-a;^ - Irx'' + 105?-) 2: 
(^6^2 (2:) = (a;6_(/2 _ loxy + 45it;y - 15^' - 180ra?/ + 120r«^(/ 
- 12ra''?/ - 45 (1 + 4r2) + 15 (1 + 2?---) x-" - .-c" + 15 (1 + &r'))z 
(/)«->/^'' {z) = (a;y - ^x'y + 30 (1 + 2r') a;^?/ - 45 (1 + ^r') xy ~ 30r (3 + 2f') x- 
+ 15?^ar» - 10«Y + 90?-a;'y- + 15a;^' - 45?y- - Ibrxy + 15?- (3 + 4r=)) 2 
^^^/r^ (5) = (ar'y' - Qxy - Qxy + 36 (1 + 2?-2) xy + 3«^ + 3(/* 
+ 4>Srafy + 48ray -18(1+ 4r2) - 18 (1 + 4r-^) ?/2 
- 48r (3 + 2r-) xy - Urxy + 3 (3 + 24?-^ + 8r^)) z. 
The above formulae may be immediately expressed in terms of Tpq by re- 
membering that <f)P\lri {z) = Tpq z. 
Thus from x^yz = + t/^ + 2r(/)) 
we get x''y = To, + T,, + 2rT,, 
and from <^'--\/^ {z) = (a;^^/ — ?/ — 2ra;) z 
we get = x^y - y — 2rx. 
9. Moments of a quadrant. 
From the preceding tables we may express any moments of z in sums of func- 
tions involving 0 and -v/r. Consider one term of such a sum : 
[ [ (^■^'i z(x, y)dxdy 
" ~ J -00 i « ^ ^ 2/) I J _^ ^^'-'f'? ^ ^ (a' .y) (^«C??/ 
= -[ <l>P-'yjr'2z{h,y)dy-r f (^-'-fiz {xjc) dx ..(85). 
