258 PEOFESSOR G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
The new independent variable r is to replace v ; it was defined in (l) by 
Z- 
= 9 „ 2 (V - 
and in accordance with ( 2 ) the equation to the surface of the pear is 
T = ~ eS.^- tffSf. 
1 / 2 t cos- B cos- 7 ^ 
0 ~ iJ ' “ siiF^ V ” 
For l)revity I now write 
' I'oin 
(^) 
2 2 -Pi)" I 1 
7 / = Vq- — - — T = ( 1 — 
A sri 2 
_ii 1 j 
2 t cos- /3 cos- 7 ,, , 
’■] =-Tvpv-= that 
0 f / T \ 2 1 COS" /3 ^ j, ^ 1 ^ COS" 7 /-1 0 \ 
— I = (1 — D jS), V' — . ■= (1 — 7j sec- y), 
sni-^ ^ sin-/3 ' ^ 1 —/3 sin-/3^ '' 
2 _ V A D 
I = — 1 — — 
^ siid/SV^ AjV’ 
0 1 ““ /d cos 2 <f) F-i" / T[ \ 
V' — 
1 - /S sin3 /3 
1 - 
IV,) 
1 — /3 cos 2cj) 
1-/3 
p- = 
1 - slid- d - cos^ (fy _ Aj^Fj^ n 1 
/c”- "■ siiF- j3 \1Y ~ Ap 
Therefore 
(z^- - p-) 7 ^- - 
1 — /3 cos 2<f) 
1 j _ Ai^d- 
1 -^ 
1 - 
AiV \IV 
sin /3 cos /3 cos 7 ( 1 — (1 — tj sec- sec- 7 )^ 
If we write 
G = \ {l + sec^ yS + see” y) 
7/ = (1 + sec^ yQ + sec'*^ y) + 4 ^ /5 + sec" y + see" ^ sec" y), 
this expression, wlien expanded as far as rp, becomes 
A 1 W 7 - 
sin /3 cos yS cos 7 
The arcs of the three orthogonal curves were denoted dn, dm, df in “ Harmonics,” 
where dn was the outward normal. Since in the ju’esent case we are measuring r 
inwards, the element of volume dv must be taken as — dn dm df. 
The equations (50) of “Harmonics” give 
