OP THE rNITED STATES. S97 



and rfC'M=tlie angle of tlie first reatlin;^ with the line joining 

 the centres. In the triangle rfCC, let 



dC'M=<t 



Cd =R=Radius of the circle, 



CC =e = Eccentricity of the instrument ; also 



let 2/7+1 denote any uneven number of equidistant readings, 

 into which the circumference has been divided, (as for even 

 numbers, the demonstration is evidently made by the cor- 

 rection of two opposite readinsrs.) and put .3=the constant an- 

 gle between tlie readings : Then we have dCM — rfC'M= 



C'dC, and sin. rf= ^jsin.'? for the first reading, or that nearest 



to the line C'CM. 



The second reading will give, 



Sin. d'==^sin.('?-f-*3). 



The third, 



Sin. d"~sin.{<p+2f). 



And the 2n"' or last reading will be, 



Sio.d"'" = ^s\n.h+2m). 

 ft/ 



The sum of all the corrections will therefore give the fol- 

 lowing series. 



Sin. rf+sin. d'+s'm. d"+ . . . +sin. d = 



„ sin. »-i-sin.(*+^)+sin.)»+2/')+ . . . +sin,(»-f-2«^) | 



The sum of the series in the parentheses is equal to 

 Cos.r.— 4^)— cos.(*-f-^*) 



