198 
MR. a. H. DARWIN - ON THE STRESSES 
Table III.—The coefficients for expressing the stress T. 
i 
A 
A 
A 
A 
A 
A 
^4 
A 
2 
+tV 
+ ■3158 
4 
J_ 2 6 0 
' 5 1 
+ 5-0980 
±80 
'17 
+ 4-7059 
_2 5 6 
5 1 
-5-0196 
6 
+14 
+ 14-0000 
_5 6 
1 1 
-5-0909 
_ 10 0 8 
5 5 
-18-3273 
_768 
5 5 
-13-9636 
±1021 
' 5 5 
+ 18-6182 
8 
4.4393 
_ 1 3248 
16 3 
-81-2761 
_ 1 0 3 6 8 
16 3 
-63-6074 
1244224 
1 5 7 0 5 
+ 42-8088 
3 0 7 2 0 
j 1 2 2 8 8 0 
_4 9 15 2 
114 1 
-43-0780 
' 16 3 
+26-9448 
114 1 
-26-9238 
f 114 1 
+ 107-6950 
10 
i 1 O 6 7 n 
7 4 8 0 0 
| 1 7 « n n 
+677280 
3 2 0 0 0 
12 5 6 0 (+0 
' 7 2 0 
+ 351-1660 
_10 2 4 00 
2 4 3 
-421*3992 
14 0 9 6 0 0 
— 5 10 3 
+ 80-2664 
1 + 43 
+ 43-9095 
2 4 3 
-307-8189 
» 2 4 3’ 
+ 72-4280 
• 17 0 1 
+ 339-3769 
7 2 9 
-43-8958 
12 
-i- 7 3 3 2 
~ 113 
+ 64-8850 
9 0 4 8 0 
113 
-800-7089 
1137280 
■ 113 
+ 1214-8673 
199840 
' 113 
+ 883*5398 
_ 8 0 6 4 0 
12 4 3 
-64-8753 
1 1075200 
' 12 4 3 
+ 865-0040 
_2 5 8_0 4 8 0 
12 4 3 
-2076-0097 
1 1474560 
' 12 4 3 
+ 1186-2912 
If W be a 2nd, 4th, or 6th harmonic these tables give the complete expressions for 
P, It, and T ; if W be an 8th harmonic the only further coefficients required are A s 
and O s . 
For the cases of the 10th and 12th harmonics the values in the tables are sufficient 
to give the stresses approximately over a wide equatorial belt, because the series for 
P, Pi, T proceed by powers of the tangent of the latitude, and the omitted terms 
involve high powers of that tangent. It would hardly be safe however to apply the 
formula—at least as regards the 12th harmonic—for latitudes greater than 15°, 
because the coefficients are large. 
§ 3. On the direction and magnitude of the 'principal stresses in a strained elastic solid. 
Let P, Q, Pi, S, T, U specify the stresses in a homogeneously stressed and strained 
elastic solid. Let l, m, n be the direction cosines of a principal stress axis. 
The consideration, that at the extremity of a principal axis the normal to the stress 
quadric is coincident with the radius vector, gives the equations 
(P—X)?+Um+Tw=0 
UI+ (Q—X)m+Sw- = 0 
TZ+Sm+ (It— \)n= 0 
These equations lead to the discriminating cubic for the determination of X, and the 
solution for Z, m, n is then 
l z m 2 _ u 2 
(Q_A)(E-X)-P 2= (P - X) (Pi -X) —T 3 (P — A.) (Q—A) — U 3 
