314 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
which they represent. The theory of elliptic integrals was developed by a method 
the inverse of that pursued in establishing the formulae of common trigonometry. In 
the latter case, the geometrical type was given — the circle — to determine the alge- 
braical relations of its arcs. In the theory of elliptic integrals, the relations of the 
arcs of unknown curves are given, to determine the curves themselves. This is 
briefly the object of the present paper. 
The true geometrical basis of this theory would doubtless long since have been 
developed, had not geometers sought to discover the types of those functions among 
plane curves. They were beguiled into this course by observing, that in one case — 
that of the second order — the representative curve is obviously a plane ellipse. Hence 
they were led by a seeming analogy to search for the types of the other integrals 
among plane curves also. 
The author hopes in a future communication to the Royal Society, the present 
having grown under his hands beyond the limits he anticipated, to extend his re- 
searches to elliptic integrals with imaginary parameters, and to show the true 
geometrical meaning of such expressions. It has long been known, that, by the aid 
of the imaginary transformation sin 1 tan-v//, we may pass from, the loga- 
rithmic to the circular type, and conversely; but it has not, however, been observed 
that this transformation enables us to effect this transition, because it changes the 
algebraic expression for the arc of a parabola into that for a circular arc or area, 
and conversely. The striking analogies developed between the formulse of the 
trigonometry of the circle and that of the parabola will be found very curious and 
instructive. 
I have attempted thus to place on its true geometrical basis, a somewhat abstruse 
department of analysis, and to clear up the elementary notions from which it may, 
with the utmost simplicity, be developed. It is only in the maturity of a science, 
that the relations which bind together its cardinal ideas become simplified. An 
author, who has himself contributed raueh to the progress of mathematical science, 
well observes, — “qui il est bien rare qu’une theorie sorte sous sa forme la plus simple 
des mains de son premier auteur. Nous pensons qu’on sert peut-etre plus encore la 
science en simplifiant, de la sorte, des theories deja connues, qu’en I’enrichissant de 
theories nouvelles, et c’est la un sujet auquel on ne saurait s’appliquer avec trop de 
soil!.” — Gergonne, Annales des Mathdmatiques, tom. xix. p. 338. 
11. I have ventured to make some alterations in the established notation of elliptic 
integrals. I have written i for the modulus, instead of c; andj for its complement 
instead of & ; so that 
The symbol c, used by writers on this subject to designate the modulus, was 
adopted by analogy from the formula for the rectification of a plane elliptic arc by 
an integral of the second order. Although in the circular forms of the third order 
it still signifies a certain ellipticity, yet it has no longer the same signification in tlie 
usual form of the first order, or in the logarithmic form of the third. 
