The Motion and Distribution of the Sun-spots 57 



and the actual values by 1^ M and L -\- ^L . The problem now is to determine 

 the corrections AJ and AL from the motions of the spots. Suppose the observed 

 co-ordinates of a spot to be ßj, one day, and , the following day. A change 

 in the position of the rotation axis will then cause a change in the latitude of the 

 spot, which is obtained from the easily deduced formula 



ß — ßi = AL sin /cos Xj — A/sin Xj . 



For the following day we obtain 



ß — ß., = AL sin / cos X., — A7 sin X,, . 



We thus leave the proper motion of the spot out of consideration and assume the 

 actual latitude of tlie spot to remain unchanged from one day to the other. Sub- 

 tracting the equations we get 



ß^, — ßi = — AL sin /(cos Xg — cos XJ -j- A/(sin X^ — sin X,) . 



Now we introduce the notation 



[p = AL sin /; 



(27) 



A/, 



and obtain after a short reduction 



(28) ß., — ßi = 2sin v{p sin u + q cos u) , 



where 



( 2u = X, -f Xj ; 



(29) 



I 2i' = X„ - 



Since X^ and Xg designate the longitudes of the spot on two consecutive days, 2v is 

 nothing but the daily rate of rotation of the sun at tlie latitude in question. We 

 have now taken all the latitude-zones togetiier and so we may appi'oximately ])nt 

 2v = 140.4 or 



V —- r.2 . 



Instead of employing , which quantity represents the mean of the longitudes on 

 two consecutive days, we refer the position of the spot to tlie central meridian 

 and write 



(30) u=-.u^J^l 



where is the longitude of the centre of the sun's visible disk, and I the longi- 

 tude of the spot referred to the central meridian. 



Now we have only included in our investigation sitch spots as are situated 

 within 60° from the central meridian. Since I is continually changing while the 

 spot is moving across the sun's surface, ß., — ß, will also be changed. We will there- 

 fore compute a mean value of ß^ — ßj. This computation may with sufficient accuracy 

 be made in the following manner. 



Lunds Universitets Årsskrift. N. F. Afci. 2. Bd 10. 8 



