X598 SDllYEY OF THE COAST 



the numerator of which is 0, for (2n+i)s=2„=ihG circum- 

 ference of the circle; and which shows that the indiscrimi- 

 nate mean point indicated by all the readings will 2;ive the 

 true ang;le at the centre of the division between any given 

 point and the line of centres. 



For any other series of readings making with the line of 

 centres an angle=<}.', the series of corrections would be simi- 

 lar to the above : 



Sin, .c+sin. /+sin, <»"+ . . . +sin.,f = 

 7- sin. 9'+s\n.(f'+i^)+s\n.{f'+2$)+ . . . +sin.('f'+27i^) 



The correction of the angle measured by these two series 

 of readings will be equal to the sum of the difference of the 

 two corrections for eacli microscope. Or, 



=(s\n.d — sin. <r)+(sin.rf' — sin. y)+ . . . +(sin. d — sin. <r ) 



=:^ sin. <p — sin. ^'+sin.(f+^) — sin.(*'+/2)+sin.(9-f-3;e) — 



sin.(<f' -1-2/2)+ . . . +s'm.(<i'+2m) — sin.(9'+2w/3), 



observing that sin. <? — sin.<i''=2sin.f(* — <f')cos.|(«+;^), and so 



on for the other arcs, the second terra, or the sum of the 

 corrections will become, 



S=^sin.|(<f'— *)[ cos. ^ +cos,(^+^)+cos.(-^+2/3) + 



cos.(^+3^)-^ . . . +cos.(^+2w^) 1 



The series in the parentheses being that of cosines of arcs 

 in arithmetical progression, is 



_ «■"■(— + -^0-sm.(—-^) 

 2sin.i^ 



