26 



to them the difference of the vahies of z corresponding to those 

 extremities, or for B, 



tzsR, + e [sin. {mn' B) — sin. (mm') ] 



but 



arc. tnn'B = a.(«— 1 ) — arc- mA, 

 arc. mn' = a.(n — 1) — (arc. mA-\-R„). 



Let g be the angular distance of Cm from zero, then 



arc Am =z j — R, 



and as R = R„ nearly, 



6 = R^ + e[sm. (a.(n— l)+B—{) —sin.(a.(n— !)—{)] 



At the other microscopes the coefficient of a is diminished by suc- 

 cessive units, and therefore making it = w and integrating, we have 

 the sum or 



„6=R+R. +Rn +2e sm.—x. 2. ^ cos. («« + — - j)| 



The integral is 



. /R , 2«— 1 \ 

 = sm.( --,+-^— «.) ^ 



2 sin. ^a 



Determining the constant, so that the expression may vanish when 

 n = 0, this becomes 



na 



sin. — - 



/R , n— 1 \^ 2 



2 = cos. (^-g- — « H 2~ "' ) 



a 

 sm.— 



This vanishes if na=2'r, and then we have 



* = i-[/J+ if.,. .. + «„] 

 or excentricity disappears from the mean of n microscopes when 

 their distance from each other is the quotient of the circumference 



by their number. 



Had z varied as any algebraic function of the sine or cosine of R, 

 it must have similarly disappeared from the mean; for it could be 



