ELLIPTIC AND HYPERBOLIC SECTORS. 445 



Suppose now m 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 f -s^) , therefore, m being a very large number, 



o^=5-T(^)andT(^)=-^: 

 2?ra 2 \lm) \2mj ma 



/", 1 a \"' r 2 a 



\am\]-.am 



we have now »•=< — \ — — / =\J- + 



6-1. M 



a m ) \ a m 



now the space a, 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 



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

 have 



1 2a J »» 



I m abi 



2 a 



16. Let us now assimie that r=M ; and since 2 a and a h are finite spaces, 



m.ao 



2 a 



and ?re is a large number, n must be a small fraction. We have now »?= -^ — r, and 



1 , 2« 



r — (\-\-nYa>>= \ (I + m)" |-«*. 



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

 2 o=a 6 ; then e wiU be a definite nimiber, which may be found from this expres- 

 sion, 



e=0- + nf. 



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

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



^+1=^.: T 



a 



1 7. The co-ordinates of the sector a being a: and y, let the co-ordinates of 

 another sector a„ be x,, and i/„ so that 



X y, m, 



a 6 



