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

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Notes:

1. Adrien Marie Legendre (1752¨C1833), Trait¨¦ des fonctions elliptiques et des int¨¦grals eul¨¦riennes avec des tables pour en faciliter le calcul num¨¦rique (Paris: Huzard-Courcier, 1825), with supplements printed in 1828.

2. Niels Henrik Abel (1802¨C1829), ¡®Remarques sur quelques propri¨¦t¨¦s g¨¦n¨¦rales d¡¯une certaine sorte de fonctions transcendantes¡¯, August Leopold Crelle (1780¨C1855), German mathematician, ed., Journal f¨¹r die Reine und angewandte Mathematik, v. 3 no. 4, 1828, pp. 313¨C323.