DUE TO THE WEIGHT OP CONTINENTS. 
193 
d?W 
dxdz 
=xz[— 2(i— 2)/3 3 cc*“ 4 +4(£ — 4 )y 8 4 2 f~ 6 2 3 — 6 (i—6)/3 Q x i ~ 8 z 4i -{- . . .] 
d?W 
= 0 , 
d*W 
: 0 
dxdy dydz 
Also treating p as identical with x, and putting y=0, 
- * (16) 
x^ = ifi Q x { —(i—2)fi s x i V+(i—4)/3 4 a? V'— . . 
dW 
y 
:0 
dy 
dW 
z—-—2fi 2 x l % 2 +4/3 4 2f~ 4 2 4 — 6/? 6 of G 2 6 + . . . 
, . . (17) 
V 
dIF 
dW\ 
cfo? n 
Vx dz) 
rflF . 
dW\ 
<& +a T y j 
: { (i/3 0 —2/3 2 )ft f 3 —[ (i —2)/3 2 —4/3Jaf % 3 
+[(i—4)/3 L —6/3 0 >/-¥— . ..} » 
/ dlF dtF\ 
= 0, 
' dIF , dW\ „ 
2/—+^— ) = 0 
dz dy 
■ (18) 
These various results have now to be introduced into the expressions (11) for 
P, Q, lb S, T, U. 
In performing these operations it will be convenient to write J for i(i+2)/(i— l). 
Also r 3 ==p 3 -f- 2 3 =£c 3 + 2 3 , when y=Q. 
From these formulas we see that S=0, U=0; which shows that a meridional 
plane is one of the three principal planes, a result already observed from principles of 
symmetry. 
Now 
, d?W 
do? 
■i{i— l)fi Q xf-\-[i(i—l)l3 Q — (i— 2)(z—3)/3J.x 2 ' V 
—[(* — 2) (i—3)(3. 2 —(i— 4)(i— 
v A -^r= i P ( p^+[}Po —(» — 2)A]af“'V-[(t—2)&-(i-4)0ja? V+ 
^ 2 IF 
(fe 2 
= -1.2^ : -[l.2^ 2 -3.4y8 4 ]^-V+[3.4^-5.6^ 6 ]^-% 4 - . 
(19) 
J 
• 2x-p^ + i 1F= — i^ Q x i +(i—4 ){3 2 x i V—(i—8)/3 4 a| % 4 + . . . 
Cvdtj 
~^iW=i/S Q o(f—if3 2 x i ~~ 2 z 2 -\-i/3 4 ;x; l ~ i z 4> — . . . 
J 
■2z^~-\-iW==ij3 0 x i —(i— 4)/3 2 a3* 2 2 3 + (/— 8)/3 4 cP V’— . . . 
2 c 
r 
• « 
( 20 ) 
MDCCCLXXXII, 
