TALBOT2999

Dear Sir,

The property to which I referred of the arcs of the hyperbola is in Chap. X. of the Trait¨¦ des Fonctions Elliptiques of M Legendre; ¹ but I do not advance that this is identical with that which you offer.

Abel¡¯s theorem ² is to be found p. 313 V.3 of Crelle¡¯s Journal, which affords the means of obtaining many results, similar to your theorem, and is I think worthy of your attention.

Your investigation of the property of the integrals under consideration shall I think embrace the determination of the constant and also the case when the hyperbola merges into two straight lines at right angles.

Making A = 1 as Legendre seems to do. The integral is

Eqⁿ 2 of M Legendres 2 Supp^t p 163: may be written thus
Whence

I find ¦Ðx = pqr
but there is a difficulty with respect to the constant C.

Your equations however for pr, qr, pq are identical with those which are given by the preceding equation making x = p, x = q, & x = r successively.

putting for a₁, c₀ & c₁ their values

which is your equation.

Are not P, Q & R each infinite?

I am, dear Sir Yours faithfully
J W Lubbock