1^ Mr. Herschel on a remarkable Application 



/2) /2) 



^' tor.3(o "= (^ ....... {34} 



r(2) r(*) , , 



>> >) 



^- /I) ',{0"^ ii-Wii-c^y l^'^r 



Let us now, instead of taking equal angles round the first 

 focus, take a series of eccentric anomalies, in arithmetical 

 progression 



, 2ir , 2(n— i)v 



ta-; OT = 73--f--— ; 7zr = ra--t- : , 



n n 



12 1 n I 



and let the resulting values of r be, as before 



^<'^r(■^ i^'\ 



I 2 n 



we get then, by the same process, 



' . CO 



- I— 2X'' . COS. nCT+^*" 



.(0 r('> ^ 



^^' r' = a. ^^..p . . .|4,iJ 



This gives, in the five cases ; 1st, when n is any number, and 



cr = 0, (or one of the r'^ terminates in the first vertex); sdly, 

 I 



when n = (sw + 1) . ^; 3dly, n = {^m + 1) . ^; 4thly, 



« = ( 6m + 1 ) . ^ ; and ^thly, « = ( 6/w + 2 ) . ^ ; the fol- 

 lowing equations 



• a \ » 



(1) jTi) n (X*— ^ -/ f , 



1- ^^ ^ =^ . \. ..-^v. ^4,2] 



(x+^-«/' 



I \* 



a at) 



2. r r — a 



I n 



