From Mary Somerville   [3 March 1831]

My dear Sir

I have great doubts as to the integration of the equation $$\frac{d^{2}r\delta r}{d{t}^2}+n^{2}r\delta r=2\int dR+r\left(\frac{dR}{dr}\right)$$ and am persuaded that in such a dilemma you will forgive me for for[sic] asking your advice. The method I have followed is that of Pontecoulant by which terms of the form $$m\mathrm{\&\#x2032;}De\cdot nt\cdot sin(nt+ϵ-\omega )$$ are avoided altogether as you will see in page [blank] but in talking the matter over with Mr Lubbock he says Pontecoulant is wrong. Now I know that the true integral is $$\begin{array}{c}\hfill \frac{r\delta r}{a^2}=2m\mathrm{\&\#x2032;}ag+\&.+m\mathrm{\&\#x2032;}fecos\{(1+c)nt+ϵ-\omega \}\hfill \\ \hfill +m\mathrm{\&\#x2032;}f\mathrm{\&\#x2032;}e\mathrm{\&\#x2032;}cos\{(1+c\mathrm{\&\#x2032;})nt+ϵ-\omega \}+\&c\&c\hfill \end{array}$$ or resolving the cosines

$$\begin{array}{c}\hfill \frac{r\delta r}{a^2}=2m\mathrm{\&\#x2032;}ag+\&.+m\mathrm{\&\#x2032;}fecos(nt+ϵ-\omega )+m\mathrm{\&\#x2032;}f\mathrm{\&\#x2032;}e\mathrm{\&\#x2032;}cos(nt+ϵ-\omega \mathrm{\&\#x2032;})\hfill \\ \hfill -m\mathrm{\&\#x2032;}fecnt\cdot sin(nt+ϵ-\omega )-m\mathrm{\&\#x2032;}f\mathrm{\&\#x2032;}e\mathrm{\&\#x2032;}c\mathrm{\&\#x2032;}nt\cdot sin(nt+ϵ-\omega \mathrm{\&\#x2032;})+\&c\&c.\hfill \end{array}$$

which is the same with La Place’s but then I do not know how I am to get rid of $c$ and $c\mathrm{\&\#x2032;}$ so that the terms containing the arc $nt$ may vanish. This point is so difficult that I trust it will be some excuse for intruding on your time.

I congratulate you on the appearance of one of the most perfect works that our language can boast of. Sir James Mackintosh begged of me to make him acquainted with you that he might express his profound admiration of your book, he says nothing has been written like it since Bacon’s nouveau anganum. Mr Hallam the historian expressed the same opinion. — We lament seeing so little of you and wish Mrs Herschel and you would spend a day with us the first time you come to town. I trust the baby is well with every kind regard to Mrs H and you believe me

very truly y[ou]rs

M. Somerville

R. H. Chelsea
^d March

Mrs S’s scruples to intrude upon you, & to stop the press in the hope that you will accede to her solicitation. To save you as much trouble as I can I send envelopes to return her sheets addressed to her, under cover to [illegible] &c.

yours very faithfully

W Somerville

The address to her is enough on the inner cover.]