ELLIPTIC AND HYPERBOLIC SECTORS. 445 



Suppose now wi to be a large number, then, the sector -^ will be small ; and its 



area will have to the area of the triangle whose base is the semidiameter, and 

 height the tangent of the sector almost a ratio of equality : now, the area of this 



triangle is ^ • T {-k~\ , therefore, m being a very large number, 



o^=^T(^)andT(^)=^: 

 2 m 2 \2m/ \2m/ ma 



2 a V» 



a m ' 



ib + 

 - \ a m [ ) ^ a m 



we have now r= { — ^ 1 =\ ^ + — i — r 



am] \ am) 



now the space «, viz. the area of the sector, is a finite magnitude, and m is by hy- 



1 a 



pothesis a great number : therefore, the lineal quantity - • — is small, and m^y 



(X Tit 



be neglected in respect of the finite line h, and m being increased continually, we 

 have 



2 a 1 »» 



I m a ) 



16. Let us now assume that -=n ; and since 2 « and a b are finite spaces, 



and 7» is a large number, n must be a small fraction. We have now m= — ^, and 



"^ nao 



1 ^ 2a 



6 



' = (l + nyab=: J (l + n)» [•« 



Now, considering that r is a function of «, let e be the value of r when 

 2a=ab; then e will be a definite number, which may be found from this expres- 

 sion, 





2^ 



To abridge, let us put n=^"*, then r„ will also be a definite number ; and we 

 shall have r=r1, that is, restoring the quantity denoted by r. 



X y a 



a o 



] 7. The co-ordinates of the sector « being x and ?/, let the co-ordinates of 

 another sector a^ be cc,, and 7/„ so that 



X y. a, 



—'-!-— = •>• 



a 



