416 ANNALS NEW YORK ACADEMY OF SCIENCES 



of either diameter by itself. Again, when both diameters are measured 

 on the same day, both are affected by the errors due to atmospheric condi- 

 tions and to instrumental peculiarities. Measures of the differences of the 

 diameters are, therefore, to a large extent, free from the troubles which 

 mark the measures of each diameter itself, and such measures furnish the 

 best test for possible variations in the sun. 



For such an investigation the observations of Schur and Ambronn 

 provide material, for on each day of observation during twelve years, they 

 measured both equatorial and polar diameters. The transit of Venus 

 heliometer measures, as reduced by Auwers, are not so satisfactory; for 

 the polar and equatorial measures were not made on the same days. In 

 each series used by Auwers, the measures of the respective diameters were 

 scattered irregularly, only a few days, however, separating the individual 

 measures; and the mean results should be more free from the masking 

 errors due to atmospheric conditions than simple determinations of either 

 diameter. Such measures should be free from instrumental peculiarities 

 and the effects of different colored shade-glasses. 



Fluctuations Having the Same Period as the Sun-spots. — The 

 writer proposes to investigate possible variations in the sun, using the differ- 

 ence of diameters (polar-equatorial) as given in the works of Auwers and 

 Ambronn. The method of equations of condition as elaborated by New- 

 comb will be used; and for the purpose of forming such equations it will be 

 assumed that the difference between the diameters (P.-E.) fluctuates har- 

 monically in a period of 11.13 years. This is the mean sun-spot period 

 as determined by Newcomb, and if there be any variation in the shape of 

 the sun, it "is just as likely to follow this rigorous period as to follow any 

 arbitrary system of numbers based upon the actual number and size of 

 spots visible at a given time. 



This assumption may be represented by an equation of the form 



P-E=pcos (jit + C)+z 



where fx is so taken that the angle ;i t -f- C shall increase by 360° in the sun- 

 spot period of 11.13 years. When the year is the unit of time, this gives 



/£=32°.25. 



The constant C determines the phase at the epoch from which it is measured. 

 This epoch is arbitrary, and in the present investigation will be taken as 

 1889.12, corresponding to a sun-spot minimum. 



Expanding the cosine term, the above expression can be put into the 

 form 



n= X cos jut+y sin fit + z 



