194 
MR. G. H. DARWIN ON THE STRESSES 
,72 t y 
+[4(£ — 4)/3 4 — 6 (i — 6 )/3 0 ] x*~ 6 z G — . . .} 
Then multiplying (19) by —({+3), (20) by 3, and (15) by Ja 3 , and adding them 
each to each, we get the expressions for P, Q, It. 
Also multiplying (21) by —(f+3), (18) by —3, and (16) by Ja 2 and adding, we get 
the expression for T. The results are 
TP = — [(T+ 3) i (i— 1) + 3 i’]/3 0 x i 
+ M+W —2 )(i —3) + 3(i—4) j/T—(f~f~3)^(^ l)/3 0 ]af 2 z 2 
-[{(*+3)(»-4)(t-5) + 3 (i-8)}fo-(i+3)(i-2)(i-3)0J^ 
+[{t+3)(f—6)(t—7)+3(i—12)}^ 6 —— i)(i — 5)/3 4 ] x l h b — . . . 
+ Ja 2 [i (z — 1) /3 0 x { ~ 2 — (i —2) (i —3)/d 2 ai*“‘V-f ({—4)(z‘—5)— . . .] 
TP=[ (i -f- 3). 1. 2/3^ -|- 3^ 0 ] x*—[(i -f - 3). 3.4^ — {(i -f- 3). 1.2 — 3 (i —4) j '"'z 2 
-}-[(iffi3).5.6/3 G — ((t-J-3).3.4 — 3 (i —8)]ht 
-[(t+3).7.8/3 8 -{(^+3).5.6-3(^-12) i G G > z '-% 6 + . . . 
TQ = — [(i+3)t—3+[ {( v i+ 3)(i — 2) — 3i}& (t+3)t/3 0 ] x‘ 2 z~ 
—[{( l *+3) (i — 4) — 3 ^} /3 4 — (i +3) (i—2 )/3j x ?- % 4 
4 - [{(^4 _ 3)(^—6) — 3 i}fi 6 — (i-f - 3)('i 4)/3 J ,]A < G z r> . . . 
+ fTa 2 [^V _2 — (i — 2)/3. 2 x t ~ i z 2j r(i — A)f3. l x t ~ G z i — . . .] 
—[{(^T‘3)2(i 2) + 3.2}/3 3 Si^o]x z 
—[{(?'+3) 4(t—4) + 3.4} /3 4 — {.(t+3) 2(i — 2) + 3 (7—2)} /3J af“% 2 
+[{0 : +3)G(^—6)H-3.6}/3 G —{(z+3)4(^—4)4-3(T—4)}/3J^“ 6 ^ 
—[{(T+3)8(^—8)+3.8}y8 s —{(T+^6(^~6) + 3(z—6)}/3 6 ]^-% 6 + . , . 
-Ja^2(i—2)L^-i(i-4:)P 4 p^+6(i-e)P^^-- . . .] 
The general law of formation of the successive coefficients is obvious, and it is easy 
to write down the general term in each of the eight series involved in these four 
expressions j the best way indeed of obtaining the formulas given below is to write 
down and transform the general term. 
The semi-polar coordinates used hitherto are not so convenient as true polar 
coordinates; I therefore substitute r, radius vector, and /, latitude, for the x, z 
system, and putting x—r cos l, z—r sin l write 
