I’KUFESSUi; G. II. DAKWIX GX THE rEAK-SllAl’ED FIGURE 
321 
l>ut K < 1, hence the tir«t sign holds true and 
M'hence 1 a/ < K/. 
Thus it follows that order i 
IV < Kd < IV < Kd . . . . < IV < Kf . 
The order of magnitude of these coefficients is therefore completely determined. 
iVs confirmatory of the correctness of this I'esult it may be mentioned that 1 lind 
that when y = 69° 50' and k = sin 73° 56', 
1^3^ = T765, K./ = -2990, = ‘4467, K.- = '4550, = -5719, = -5876. 
When y -= 75° and k = sin 81° 4' (another Jacobian ellipsoid) the numbers run 
•130, -224, -460, *465, '604, -614. 
We see that tor the liarmonics of higher order the ellipsoid is more stable than it 
was and for those of lower order less stable. 
§ 7. The critical Jacobian Ellipsoid. 
From a number of preliminary calculations 1 saw reason to believe that the critical 
ellipsoid would be found within the region comprised l)etween y = 69° 48' and 
69° 50', and sin“^ k = 73° 52' and 73° 56'. 
If Ave write 
,V ■ -I N T /h , «^Bhr7C0,s~7\ /cViu7cos7 (1 + «:Vin-7) 
J iy, sm ^ k) = 7., 1 + -- . 0 - — ( 2 F — It)-777-- o • - n- 7 : -’ 
^' ’ K ~ \ 1 — Sin- 7 / ' ' /c - (1 — K- sun 7 )’ 
w liere the amplitudes of E and F are y and their moduli k, tlie existence of the 
Jacobian ellipsoid is determined by 
f{y, sin ^ k) = 0.-" 
The coefficient of stability is 
- P.'eO Q.'U') 
The form\dce for computing liitg are given in ^ 4. 
The values of E and F are from Legendre's tables. 
* See ‘ Roy. Sue. Froc.,’ vul. 11, p. 323, where the foi'iiiuhi is reduced to ;i i'orni eonveiiiciit for 
computHtion. 
