3-15-18. George Howard Darwin to H. Poincaré
Aug. 3. 1901
Dear Monsieur Poincaré,
Since I wrote last I see that the method of my last letter gives less than I hoped for.11See Darwin to Poincaré, 31.07.1901 (§ 3-15-17). This I will explain. If
is the first approx. to the surface under rotation , the problem we should like to solve is to find the next approx.
Suppose the next approx. to be
where and are at least of order . This is to be solved by making stationary, where is lost energy & m[oment] of i[nertia].22The total energy of the system is , or in Poincaré’s terms, . If we concentrate the layers represented by and we see that their contributions to are of order and and are negligible ex hypothesi; their contributions to are nil as being harmonics of orders 4 & 6. Hence we may as well start by omitting and . (I had in fact gone thro’ the work and found that they do disappear entirely before I realised the reason of it).
Suppose then the surface to be
where the is not the same as the of my last letter. But we are supposed to be ignorant of the nature of the true figure of equilibrium and therefore we must merely regard as a parameter whose cube is to be retained.
If we write as the mass of the body, it is easy to show that
a result needed later.
Proceeding as in my last letter,33Darwin to Poincaré (§ 3-15-17).
Concentrate the layer of neg. density on the sphere & we get surface density as far as equal to . Express this in harmonics retaining only harmonics whose coefficients are of order but developing those coefficients as far as (you will perceive that this is sufficient) & we get surface density
Find the lost energy of this by the usual method & we get
Exactly as in my last the energy lost in distorting the layer is
The energy lost in expanding the layer is
Adding together (2), (3), (4), (5), (6)
Introducing from (1)
The moment of inertia is only counted as far as . For the sphere
For the layer
From (8) and (9)
From (7) and (10)
Making this stationary for variations of
This is all that is attainable from this method, but I want to show that it is right & for that purpose I must find what really means when we know that the resulting figure is an ellipsoid. If I write for the of my last letter, I showed that the equation to an ellipsoid was44In Darwin to Poincaré, 31.07.1901 (§ 3-15-17), denoted half the square of ellipsoid eccentricity; see equation (A).
If I put
|this may be written|
and this is the form with which I have worked.
If is the eccentricity of ellipsoid we had . Hence
Now I have proved (tho’ I cannot refer you to any book for the result) that
Taking the first two terms of this series we have55Cf. Thomson & Tait’s formula (1879, § 771).
Thus the result is correct.
It appears however that unless the approximation can be carried as far as there is no way of determining what meaning is to be attributed to .
In looking over the investigation, I see that , , and can be found for one more power of ,66Variant: “can be found as far as ”. but I do not see how and can be found more exactly.
I conclude from this that the application of the similar method to the pear would throw no light on the further approximation to its figure, but would enable us to determine whether or not the pear corresponds to greater or less m[oment] of m[omentum] and so absolutely determine the question of stability. I am not sure whether or not I shall have the patience to carry out the enormous labour of the investigation. I wish I could see my way to more accurate determination of the figure.
The impasse in which I find myself is quite unexpected by me, although you very probably foresaw it.
I feel quite ashamed to have troubled you by these enormous letters, and I doubt whether you will have the patience to read them.
I remain, Yours very truly,
G. H. Darwin
ALS 6p. Collection particulière, Paris 75017.
Time-stamp: "18.09.2016 01:33"
- Treatise on Natural Philosophy. 2d edition, Cambridge University Press, Cambridge. External Links: Cited by: 3-15-18. George Howard Darwin to H. Poincaré.