72 



O. A. Åkesson 



the Dieau of the frequency-surface as the origin of a system of co- 

 ordinates, with the axes parallel and perpendicular to the equator, the curves of equal 

 frequency will be represented by the ellipses 



— xy -f = constant. 



^20 ^20^02 ^02 



Here x and y designate the co-ordinates in this system. In order to make the axes 

 of our system of co-ordinates coincide with the principal axes, the system must 

 be turned an angle 'f, which may be easily computed in the following way. Suppose 

 X and Y to be co-ordinates referred to the principal axes, we have 



(39) 



{X = x cos + y sin 'f , 

 \Y=y cos cp — X sin . 

 Since X and Y are uncorrelated, 



Ï.XY=0. 



Hence it follows, by multiplying the equations (39) and summing, 

 0 = (— ^20 + V02) sin 2'f + 2v^i cos 2'f , 



or 



^20 ^11 



Since 



Oj. > a„ , 



we have 



0 < 'f < + 90", if r > 0, 



and 



0 >9 > - 90°, if r < 0. 

 The dispersions and Ï.,, about these new axes, the principal axes, are easily 

 obtained from the equations 



1 = o^o.Vl - . 



Since, in the present case, r is a small quantity, the new dispersions will differ 

 inconsiderably from o^. and Oy. The results of the numerical computations are given in 

 table XXXI. From this table the fact of the correlation coefficient r being negative in 

 the northern hemisphere, and positive in the southern, is clearly proved. The angle 

 between the line of symmetry of the frequency-surface and the equator, seems, upon 

 an average, to increase with increasing latitude. These two facts are evidently due 

 to the different angular velocities of the sun at different latitudes. Hence, the trade- 

 winds on the earth, have to some extent their correspondence on the sun though 

 their their directions are inverted. 



In this connection we will devote some remarks to the following problem: Is 

 if possible to divide our correlation-surfaees into two normal frequency-sur/aces ? Pro- 

 fessor Chaklier has in his Stellar Statistics II, page 93, completely solved the 

 problem of dividing a given correlation surface into two spherical components. 



(41) 



