2 SIXTH REPORT. — 1836. 



sum, which can only be done successfully in certain cases. In all 

 other cases it is requisite to combine three or more integrals, and 

 the idea of doing this seems not to have occurred to these illustrious 

 analysts. 



Thus, for instance, they tried in vain to find the algebraic integral 

 of the equation P ^ ^^ , f ^V _ q 



But if they had sought for the algebraic integral of 



/ ' dx r dy r I 



Vi + x^ J Vi + y* J vr 



= 0. 



they would have found that such a solution really exists. 



However, the theorem which Euler gave, although limited in its 

 extent, yet jjroved to be of great importance, and may be considered 

 the foundation of the theory of elliptic functions given by Legendre, 

 the different properties of which are implicitly contained in Euler's 

 solution, although Legendre's talents and industry were requisite 

 to draw them forth, and develop them with clearness and precision. 



While examining this subject in the year 1825, 1 met with a new 

 property of the equilateral hyperbola, which appeared to me to be of 

 great importance, as it gave the algebraic sum of three arcs of that 

 curve. 



If the abscissae of the three arcs are the roots of a cubic equation, 



of this particular form, viz. :a?' x — r=0, I found that the 



sum of the arcs was an algebraic quantity. In this equation the 

 letter r is arbitrary. Each particular value which is attributed to it 

 furnishes a solution of the problem, that is, it gives three arcs whose 

 sum is algebraic. 



I verified the truth of this theorem by numerical computations of 

 different examples of it, but in so doing I met with two difficulties 

 of a novel nature. The first was, that by attributing certain values 

 to r, the cubic equation had two impossible roots, and the theorem 

 then apparently ceased to have any real meaning. (At that time 

 Legendre had not yet demonstrated the fact, that two imaginary in- 

 tegrals can make a real integral by their addition.) 



The other difficulty was this, that in making the addition of the 

 three integrals, I found that it was necessary to attribute a negative 

 sign to one of them, and although by making actual trial in each 

 numerical example, it was easy to see which of the integrals had 

 this sign, yet it was by no means so easy to assign a convincing 

 reason why this ought to be the case. 



The method which had conducted me to this theorem respecting 

 the equilateral hyperbola, would, as I saw, furnish a multitude of 

 other theorems equally curious ; but the field of inquiry was so new, 

 and the results which it afforded at every step so ample, that I was 

 at a loss how to classify them, or reduce them into a clear and con- 

 nected theory. For instance, I perceived that I might consider n 



