ELLIPTIC AND HYPERBOLIC SECTORS. 447 



2 <> 1 



18. We have found that - h •7=6" ' , and here «=(!+«) ", and w is a very small 



a 



fraction. This function e, from its singular form, might be supposed indetermi- 

 nate ; yet its value is readily fovmd, by expanding it by the binomial theorem, 

 to be 



^ = l + ^0 + r72^3:4 + l-2-TT:^ + ^^=2-7182818284. 



This number e is the Base of Neper's logarithms, and therefore 



^« = Nep.logg+f), and«=^*Nei,logg+|) . . . . U 



-+f) = ; Therefore also 5--|=e~«^' 



a 0/ !^_y a b 



a b 



2„ 



and hence ?=i|e«' + e "'■j, 



2a 



y , f a6 ab\. 



By a process exactly sunilar, we might have deduced, in the case of the 

 circle, supposing a=b=l, these other important formulae : 



i a=x^=^ J. 



u,J—\ — aV— 1 



e +e 



- 1 sin I 



Here a denotes, not the sector, as in the hyperbola, but the angle of which x 

 is the cosine, and ?/ the sine. But the various steps of the process would be 

 almost exactly the same as those by which the corresponding properties of the 

 hyperbola have been found. 



We have now passed, by a simple and uniform analysis, from the definitions, 

 and the most elementary geometrical properties of the conic sections, to some of 

 their most recondite properties. With these exponential formulae, which were 

 turned to great account by Euler,* and have been pronounced by Lagrange to 

 be the finest analytical discoveries of the last century, f I conclude this memoir. 



* MisceUanea Berolinensia, tome vii. ; and Introdiictio in Analysin Infinitum, t. i. 

 t Lagrange. Legons nur le Calcul des Fonctions, p. 1 14. 



VOL. XIV. PART II. 3 Y 



