OF ROTATING COMPRESSIBLE MASSES. 
187 
and the value of Q ' 0 is 
so that 
n , R' 6 3S' 4 2, 
Q 0 = -To w + -n wV+ 
a 8 c 4 
Q' = RV + 3S^> + 3T , m»? 4 +Ur + r , w 4 + 2s'w>^ + tr + 2uV + 2vr. • (132) 
Let us now assume that 
£ (Q'-i/DQ'+A/*D*Q'-Ww/*r»Q') ^ 
= h 1 (5w 6 —90w i z 2 +l20w 2 z 2 —l6z i )+h 2 (3w i —24:W 2 z 2 +8z i )+h s (w 2 —22?), . (133) 
this being the most general form possible, since the integral in question is necessarily 
a zonal harmonic. In calculating the value of DQ', we may use the transformation 
D = 4-r-^(w~)+ X 8 " 
a 2 A ts St? / c 2 C 3£ 2 ’ 
and on equating coefficients in equation (133), we obtain 
1 / X . X . 1 / X 
bh 
-r 
A 6 4A 5 C 2 A 36R + c 2 C 6S ) + 64A 4 \a 4 A 2 
576R' 
a 2 c 2 AC l928 ' + c'C* 721 -') 
1 ' X " 2304R' + , 3X 'L. 384S / + 3X " 288T' + ~ 7201R 
23 04A 3 \a° A 3 
a 4 c 2 A 2 C 
aV AC 2 ' 
^ 6 C 3 
dX 
A’ 
(134) 
3/r — 
r 1 / X -t r> ) , X j_/\ . 1 / X n/>T>/ . X 
A 4 4A 3 Va 2 A 16 r/+ c 2 C 4s ^ + 
„ A4 , , A 36R/+-^6S' 
4 A A c 2 C 
1 ( X 2 ^ , , 2X 2 , , X 2 0(4/ 
6 4 A 2 WA 2 a 2 c AC c 4 C 2 
1 / X 2 
82A3\aW 576R ' + ^C 96S ' + ^ 72T ' 
+-(~TaA 2304R 7 + - . 384S' + 3 ,J . A A ri „ 288T / + 720U / 
2304A W A 3 «VA 2 C « 2 c 4 AC 2 c b C 3 / 
.... (135) 
^-7t(-LC 8u/+ ^ 4v ') + TTT(-vr 16r/+ ^ 4s ' 
A 2 4 A WA AC / 4A 2 \a 2 A c 2 C 
3 X s 
3X 3 
X 3 
dX 
A 
*, = f 
Jo 
1 
32 A \a 4 A 
X 2 
; 64 r ' + ^c 16 s ' + ?b 24t ' 
+ 6^(^ 576E ' + ^C 96S ' + o^ 72T ') 
2304A WA 
X 3 
-2304R'+ 
3X 3 
a 4 c 2 A 2 0 
384S' + 
3X 3 
X 3 
2 288T'+ AA 5720 TJ 7 
a 2 o'AC 2 c b C 3 
dx 
A ’ 
(136) 
