WCP2098

[1]¹

14 Waverley Place

Nov. 26^th 1867

Dear Mr. Wallace,

I wish to make a remark or two on your reply to my letter of Sunday last.²

A perfectly ^uniform flexible but inextensible string, fixed at its extremities; would under the action of gravity (or what is the same a weightless string acted upon by an infinite number of vertical forces) assume the a form somewhat resembling a parabola, though the curve with which it would coincide (the catenary) has very different geometrical properties. If the string were extensible [2] and perfectly elastic the curve formed would be a modification of the catenary but not a parabola.

If however the string were weightless and acted upon by forces ^{all in one plane and} everywhere perpendicular (normal) to the elements of the string, then the form of ⁱⁿ when equilibrium was established would ^{I believe} be an arc of a circle no matter whether the string were inextensible, but longer than the distance between the fixed points a and b³ a or inextensible and perfectly elastic.

I had this in view when I stated that your membrane when submitted to ^{the uniform} pressure of a gas or liquid (which is everywhere [3] normal to the elements of the surface) would probably assume a spherical form. I was guided in making this statement by the "principle of sufficient reason"[.]⁴ I see no reason why under the given circumstances the spherical form should be departed from[.]

You allude very properly to the case where the surface might be extended by sl increasing the pressure beyond the hemisphere.⁵ if it were however I see no reason why an ellipsoidal form should be departed from assumed. The membrane would probably remain spherical as a soap bubble appears [4] to do when carefully blown for a sufficient time. Of course I am aware that the conditions in a soap bubble are not precisely those of your elastic membrane.

B When I have leisure I will look further into your problem. At present believe me to be

in haste | Yours very truly | T. Archer Hirst [signature]

P.S. When a bladder in india [sic] rubber is tied over a tube and the air exhausted it assumes an elongated form as in the adjoining figure.⁶ But here[?] your conditions are not fulfilled[,] the membrane loses its uniformity of texture[,] its thickness at the lower extremity diminishes and it is imperfectly elastic.