ELLIPTIC AND HYPERBOLIC SECTORS. 443 



From this, and the first given equation, there is obtained 



The same result as was deduced from the first solution but by a different process. 



13. In the preceding example the imaginary expression V— 1 has been elimi- 

 nated by a transformation, which has brought together two terms with oppo- 

 site signs. In a similar way we shall eliminate it from formidge (N). 



By the binomial theorem, and putting Aj, A^, A^, &c., for the coefficients of 

 the terms containing the first, second, third powers, &e. of ^ in the development of 

 {!+!/)", we have 



(z + y^c)''=x'' + A^x"-' y^c + A^z'-^ y' c + A^x'^y^ Cyfc + kc. 

 (z-y^cy = z"-AiX"-'i/Jc + A^x"--t/''c-A^x''^i/'^ c^c + kc. 

 Now from formulae (N), 



^cF ina)^i{Qx+y^cy-(x-ysfcr}. j 

 Therefore, by adding and subtracting, there is found 



/{na)=x" +A^x^''i/'c + A^x"-^y*c 



F(n 0)=: A^x'-'y + A^ x"-^y^ c + A^x"-^y^c^ 



O 



= + &C. f 



These series wiU terminate when n is a whole number, otherwise they wUl pro- 

 ceed ad infinitum. In the circle, or ellipse, c must have the sign — , but in the 

 hyperbola, the sign + . 



John Bernoulli found these theorems in the case of the circle, and gave them 

 in the Leipsic Acts for 1704, but Avithout demonstration.* It is remarkable that, 

 knowing them, he did not discover also De Moivre's theorem, which has been 

 the geiin of the finest discoveries in geometry. 



14. In the circle, tan a, the trigonometrical tangent of an angle a at its centre 



is equal to, ^^j-^. In the ellipse or hyperbola, if a straight line touch the cun^e at 



the vertex of the transverse axis, the segment of this line between the vertex and 



any semidiameter is equal to y|", ; it may therefore be considered as a function 



of the sector, analogous to that which the trigonometrical tangent is of its angle ; 

 and may be similarly designated by the symbol T («). 



* JoANNis Bernouilli, Opera, vol. i. pp. 387 and 511. 

 VOL. XVI. PART. II. 3 X 



