PROFESSOR G. H. DARWIN ON THE PEAR-SHAPED FIGURE 
O O o 
N 
O'W 
Uf > = as, 
1:3/(i^o) ^/(g.) < = > <> 
If we express tlie iQ’s in terms of integrals this becomes 
a9,-(r)|9,'WP(w-l)*(w-!-!|)‘ 
> 0 
The seventh proposition shoves that when s and t are greater than zero, and i is 
greater than 2, all the elements of the integral have the same sign. Hence the 
(juestion is whether. 
^ ^ 
Therefore Ave liave to arrano-e all the w in descendino' order of mao;nitnde, 
and shall thereby obtain the non-zonal 3 a’s in ascending order. 
I wish first to show that these coefficients may to a great extent be sorted by 
considering the inequality. 
^V (^o) ^ 
l?(ib ^ ^ Ho?') 
. 1 , 2 , : 
SnpjAOse, if possible, that Avhereas, for the ellipsoids defined by ^8, a, zf,-„ 
m^o) . Woy) , Mfo) . PfiO'o) 
IV (^) HVV ■ 
Then there must be some value of /3 for which 
5 ywww = zy«w(>'») 
for all A^alues of a greater than a,,. 
It is almost obAuous that there is no one value of /3 which renders this equation 
possible; but consider for example the case of s = 2, / = 0. 
Now 
(^) = — ^90^3 H + = P 3 + ^'7:^3^ (Gi)- 
It we substitute this in the equation Ave find 
WWP 3(0 = P,,= W 
This can only be satisfied by a = Pq, and hence the hypothesis is negatived. 
Similarly the assumption of other Axrlues of and t leads to an impossibility. 
Thus Ave may consider the P functions in place of the P functions. 
