From Horace Darwin to G. H. Darwin [February–April 1875]

Dear G

It seems to me when you were talking of inverting force by a cell you were talking bosh; the following seems to me the only way of using a cell with force wh. is possible, & it does not invert.

Let P be the poles of a cell

f a force acting at P

f′ . . . . . . . . . . . . . P′

[DIAG HERE]

oP=r

oP′ = r′ ∴ rr′ = cont.

= k suppose

Suppose the cell to move a little so that r becomes r+dr , & r′ becomes r‘+dr‘

Then by virtual velocities

fdr = f′dr′

and r‘dr = –rdr′ ∴ dr =–$\frac{\mathrm{r}}{\mathrm{r\prime }}$dr′

∴ f$\frac{\mathrm{r}}{\mathrm{r\prime }}$dr′ = f′dr′

-f$\frac{\mathrm{r}}{\mathrm{r\prime }}$ = f′

∴ –fr = f′r′

–f rr′ = f′r′²

f = –$\frac{\mathrm{f\prime r\prime 2}}{\mathrm{k}}$

Hence if f is a constant f′ varies as the square of its distance from o.

If you can invert this by another cell the thing is done; but I don’t think it would work practically.

I get a problem from this.

A body is moving in an orbit round a central force wh. varies as the square of the distance; if R & R′ be the distance of the body from the force at any two pts of the orbit, the amount of energy (kinetic & potential) lost, of gained, by the body in moving from one of these pts to the other, will vary as $\frac{1}{\mathrm{R}}$ – $\frac{1}{\mathrm{R\prime }}$

This is done at once by the cell.

Suppose we make our force varying as square of the distance by a cell, with a constant force acting at one of the poles (P) and the body at the other (P′)

Referring to old picture

work done in moving from one pt. to another

= f . Δ r = k f Δ ($\frac{1}{\mathrm{r\prime }}$)

where Δ r = change in length of r

H Darwin

[DIAG HERE]