149 



the femi-axls major i. But when the excentricity is confiderable, this le. 

 ries, converging flowly, ceafes to be ufeful. Several geometricians of the 

 firfl; rank have particularly confidered this difficulty. The refult of their re- 

 fearches being connefted with the complete folution of the above problem, 

 properly requires fome notice here. 



If X be the diftance of an ordinate from the centre of the ellipfe, the 

 fluxion of the elliptic arc intercepted between the ordinate and extremity 



of the axis minor is — -p:= x v/ i - e' «' • This fluxion is eafily re- 



p* 

 duced to the form ji=rrz== P being a rational funftion of 



^a+bx+cx '■ -j- dx ^ -]-ex ''^ 



X. The eminent mathematicians Euler, Lagrange, and Legendre, have 

 employed themfelves on this form. Lagrange has been particularly fuccefs- 

 ful, and by a mofl ingenious procefs has ftiewn that it may in every cafe be 

 transformed fo that its fluent may be obtained by fwiftly converging feries. 

 His memoir on this fubjeft is to be found in the " Mem. Acad, des Sci- 

 " en. de Turin 1784, 1785," and his method is juftly lliled by Lacroix * 

 " La plus elegante pent etre qui foit fortie de la plume des analyftes." How- 

 ever, in the cafe of the excentric ellipfe this method does not furnifh fo 

 fimple a folution as may be otherwife derived. For in the application of 

 Lagrange's method, a remarkable theorem offers itfelf, f by which the 

 circumference of one ellipfe may be derived from that of two others lefs 

 excentric. The firfl difcovery of this theorem is due to Legendre, and 

 was given, derived by a different method, in the 2d of his two very inge- 

 nious effays on elliptic arcs. | He had firfl difcovered that the circumference 

 of one ellipfe may be derived from the circumference of another ellipfe by 

 means of partial differences, and afterwards combining this conclufion with a 



method 



* Tralte du calcul. diff. et integral, vol. 2. p. 88. 

 t Lacroix. vol. 2. Art. 506, 507 

 t Mem. Acad. 1786. 



