830 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Then we should have 
G=im 
l + 2(l-X)-y(l-X)(l-2X) 
D= i 
9.^.3 
l_2(l-X)+y(l-X)(l-2X) 
The last term in each of these expressions involves a small factor both in numerator 
and denominator, viz. : 1 — 2X because nearly, and p, because the viscosity is 
large. The evaluation of these terms depends on the actual degree of viscosity, but 
all that we are now concerned with is the fact that when X = §- the true physical result 
is that D changes sign by passing through zero and not infinity, and that G does the 
same for some value of X not far removed from -}>. 
Now consider the function 
4\(1 — A) 
1 —2 A, 
The following results are not stated retro- 
O 
spectively, and when it is said that i or j increase or decrease, it is meant increase or 
decrease as t or £ increases. 
(i.) From X=1 to X= - 5 the function is negative. 
Hence for these values of X the inclination j decreases, or zero inclination is 
dynamically stable. 
When X= - 5 it is infinite ; but we have already remarked on this case. 
(ii.) From X='5 to X= - 191 it is positive. 
Therefore for these values of X the inclination j increases, or zero inclination is 
dynamically unstable. It vanishes when X=T91. 
(iii.) From X=T91 to X=0 it is negative. 
Therefore for these values of X the inclination j decreases, or zero inclination is 
dynamically stable. 
Next consider the function l + “y r—. 
(iv.) From X= 1 to X=’809 it is positive. 
Therefore for these values of X the obliquity i increases, or zero obliquity is 
dynamically unstable. It vanishes when X= - 809. 
(v.) From X=’809 to X= - 5 it is negative. 
Therefore for these values of X the obliquity i decreases, or zero obliquity is 
dynamically stable. 
When X=’5 it is infinite ; but we have already remarked on this case. 
(vi.) From X= '5 to X=0 it is positive. 
Therefore for these values of X the obliquity i increases, or zero obliquity Is 
dynamically unstable. 
