418 ANNALS NEW YORK ACADEMY OF SCIENCES 



the entire series of heliometer measures from 1873 to 1902. The results of 

 these three solutions are shown below; 



Series. x. y. z. 



1873-85 +0".255 +0''.015 +0''.156 



1890-02 +0".032 +0''.001 +0''.019 



1873-02 +0''.048 +0".006 +0".029 



The probable error for x in the whole series is ± 0".021. Thus the value 

 of X as found from the equations is slightly more than tv/ice its probable 

 error. Moreover, in each series the three quantities come out with the 

 same sign and approximately of the same relative values. Reducing the 

 results to monomials, we have finally for the three determinations 



P.-E.= +0".256 cos (jtct-4°) + 0''.156 series of 1873-85. 

 = + 0*'.032 cos 0^t-2°) +0''.019 series of 1890-02. 

 = + 0^.049 cos Oit-6°) + 0".029 series of 1873-02. 



These results were obtained by assuming a harmonic variation having 

 a period of 11.13 years. They show that the phases of such a variation co- 

 incide to within one-fifth of a year with the phases of the sun-spot fluctua- 

 tions; that, at times corresponding to minimum of sun-spottedness, the 

 polar diameter is relatively larger; that, at times of maximun sun-spotted- 

 ness, the equatorial diameter is relatively larger. 



The amplitude of the variation is extremely small, but its reality would 

 seem to be established. The present investigation at least renders the 

 existence of such periodic fluctuations in the shape of the sun more probable 

 than their non-existence. 



Search for Short-Term Periodic Variations. — If the equator of 

 the sun were of permanent elliptic shape, then we should have a periodic 

 variation in the observed differences between the polar and equatorial 

 diameters, and the period of this variation would be equal to the sun's 

 sjTiodic rotation. While it is extremely improbable that any such perma- 

 nent deformation exists, yet semi-permanent deformations may readily 

 occur. The sun-spots are local phenomena; and when large spots exist 

 on one portion of the surface, the equator may be deformed in such regions, 

 and such deformation may persist during many rotations of the sun. 



Unfortunately for investigating the question of the existence or non- 

 existence of fluctuations in the measured shape of the sun, corresponding 

 to possible deformations of the equator, the sun's synodic rotation is not a 

 well defined constant. Different portions of the surface rotate in different 

 periods. According to the latest spectroscopic researches, the equatorial 

 regions rotate in 24.46 days, and regions in latitude 80°, in 30.56 days. 



