Lacock Abbey
8 Septr 1844
Dear Sir
I shall have great pleasure in forwarding to Mrs Somerville <1> and also to yourself copies of the mathematical memoirs <2> – I think they will travel very well by post.
I had advanced the subject a good deal further than appears in those memoirs, but I never could find time to put the materials into proper order for publication.
The subject however is a great favourite of mine & I should be very glad to continue it, and equally glad if anybody else would continue it – at every [illegible deletion] step you take, you start coveys of beautiful theorems (excuse a September metaphor) which hardly by any other means can be got hold of, else Legendre <3> and others would certainly have found them. Will you give me leave to ask a matheml question or two which occur to me relative to this subject?
What is the definition of an Abelian integral? <4> for it appears to me that most integrals possess the Abelian property.
What is the definition of a Cone of the second order? I must be under a delusion of course, for I cannot see how it differs from a common oblique cone.
Is it not the same thing, presented differently to the contemplation?
Yours very truly
H. F. Talbot
Sir J Herschel
&c &c
Notes:
1. Mary Somerville, née Fairfax (1780–1872), writer on science.
2. WHFT, ‘Researches in the Integral Calculus, Part One’ Philosophical Transactions of the Royal Society of London, v.126 part 1, 1836, pp.177–215 And WHFT, ‘Researches in the Integral Calculus, Part Two’ Philosophical Transactions of the Royal Society of London.
3. Adrien Marie Legendre (1752–1833), mathematician.
4. Also called hyperelliptic integrals. This is the first known use of the mathematical term 'Abelian Integral.'