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 di£ferent 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 formulae (N). 



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

 the terms containing the first, second, third powers, &;c. of ?/ in the development of 

 (l + t/)", we have 



(x ^i/^cy^x" + A^x"-' yslc + K^x^'-'^y^ c + A^x"-^ y^ c^c + &c. 

 {x-y^cY^x^-A^x"-' y^c + A^x"-^ y^ c-A^x'^ y^ c^c + hc. 



Now from formulae (N), 



f{na)=l{(x+yjcy + {x-yjcr} 

 ^cY{noi)^\{{x^y^c) 



Therefore, by adding and subtracting, there is found 



/(na)-x" +A^x'^'y^ c + A^x''-'y'^c'^+kc. 



F{n a) = A ^x'-'y + A^ x"-^ y^ c + A^ x"-' y^ c^ + &c. 



These series will terminate when w is a whole number, otherwise they will pro- 

 ceed ad iiiflnitum. 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 without demonstration.* It is remarkable that, 

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

 the germ of the finest discoveries in geometry. 



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



QIJT ft. 



is equal to, . In the ellipse or h3^perbola, if a straight line touch the curve at 



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



any semidiameter is equal to -r. -, ; 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 





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