18 PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
the boundary density to the mean density of all inside of it; the third line gives 
Pq/P (computed from (2G)), by which we trace the variations of pressure at the 
boundary. 
Table II. 
Value of a. by reference I ® 
to former solution / Uj 
= 
0 
■1 
-2 
-3 
-4 
-5 
-6 
-6264 
-7 
-8 
_ 
-103 
r_ 1 1 
00 
1-7577 
-6741 
-4826 
-4236 
-4031 
-3974 
-3972 
-3984 
-4027 
/iMj a 
3 P 
®i 
Wf, 
r 
- i-H d^j/ddxP" 
1-0000 
-8841 
-6838 
-5333 
-4383 
-3791 
-3410 
1 
-3158 
-2986 
P 
dijjdx-^ 
3 
Pnf- 
— d~y-^ld.rp~ 
00 
4-662 
1-383 
-772 
-557 
-458 
-407 
-397 
-377 
-361 
4^1 
[dijJdx-^P _ 
Value of a by reference 1 ^ _ 
to former solution J 
-9 
1-0 
1-25 
1-5 
2-0 
2-5 
3-0 
00 
-ipsf- 1 1 
= 
-4085 
-4152 
-4325 
-449 
-476 
-487 
-497 
1 
2 
p.Mla ^ L dy-^jdx^ 
— d-t/jldxp" 
’2867 
-2785 
-2676 
-264 
-267 
-269 
-273 
1 
P L dyjdx-^ 
3 
Pnl"— — d-yJdxP' 
= 
-351 
-347 
-347 
-356 
-382 
-392 
•406 
1 
2 
J-^L {.dy-^ldxj]- _ 
The minimum value of Wq! p occurs when aja^ = 1’6 very nearly, for, when 
a/a-^ = 1'4, 1'5, 1’6, I find = •26521, ’26437, ’26425 respectively.* When 
r/a^ = 1’6, = — *38435 and dyjdx-^ = 3’5I80. The minimum value of pJP 
occurs when aja^ = T1 very nearly, for, when aja-^ = I'O, I’l, 1*2, I imd pJP = *3469 
’3455, ’3462 respectively. 
When wjp is a minimum, the density at the centre is 381 times that at the 
boundary, and, when p^jP is a minimum, the density at the centre is 129 times that at 
the boundary. M. Ritter finds the pressure to be a minimum when this ratio is 258, 
instead of 129. As this corresponds to = 1’5, this discrepancy between our 
solutions is not so large as might be expected from the great discrepancy between 
these results, and I cannot but think that my result is more accurate than his. 
The minimum value of -g-/3” occurs when aja^ — ’6264, and its value is ’39723. 
This value makes the surface density exactly one-thh’d of the mean density, for is a 
minimum when x-^ dijpdx-^ is a maximum, and this occurs when xpPyJdx^ fi- dyjdx-^ — 0; 
and, when tliis relationship is satisfied, w^jp = 
It is interesting to note that in this case /3^ is very nearly equal to so that the 
* Ml’. Hill finds tliat the minimum value of w^jp approximates to yy, or ■26G7. The agreement 
between our results is satisfactory. 
