OP ARTS AND SCIENCES. 129 



If, from the standard catalogue thus formed, accidental errors can be 

 excluded, comparing with any other catalogue, we shall have directly 

 for each star periodic equations of the form, — 



m sin a + n cos a = r, 

 m' sin 2 a + n' cos 2 a = r', 

 a sin (J -\-h cos 6 = r" ; 



from which, the periodic coefficients can be successively found by 

 least-squares ; and, with these coefficients thus obtained, a system of 

 corrections may be computed for each hour of Right Ascension. 



In the absence of any system of observations since 1858, made with 

 especial reference to freedom from periodic errors, this method will 

 be employed in this investigation. The standard catalogue selected, is 

 given by the following title : — 



" On the Right Ascensions of the Equatorial Fundamental Stars, and 

 the Corrections necessary to reduce the Right Ascensions of Different 

 Catalogues to a Mean Homogeneous System. By Simon Newcomb, 

 Professor of Mathematics, United States Navy." 



This catalogue of 32 fundamental stars proves to be far more accu- 

 rate than any hitlierto constructed. Indeed, it seems almost suj^erfluous 

 to attempt any corrections for accidental errors. I shall, however, ven- 

 ture a thorough comparison with only modern observations, on the 

 supposition that they are somewhat more accurately made than earlier 

 ones, especially those made since the introduction of the chronograph. 



Since the assumed system of Right Ascensions is supposed to be 

 entirely homogeneous, the coi'rections obtained will be independent of 

 the time of observation ; and, when once obtained, may be applied to 

 the whole system. I conceive it to be possible, in this way (assuming, 

 of course, that the annual valuations are correctly known), to correct 

 the accidental errors of any part of the system, by observations made at 

 the point of time and under the conditions most favorable to accuracy. 



A preliminary and tentative discussion, by the method above indi- 

 cated, gave the following periodic equations, viz. : — 



S. S. 8. 



Greenwich. — Observations from 1860 to 1870, r = — .010 sin a + 007 cos a. 



Edinburgh. — Observations from 1860 to 1869, r = — .009 sin a +.006 cos a. 



Oxford. — Observations from 1860 to 1870, r = — .012 sin a +.019 cos a. 



Washington. — Observations from 1860 to 1870, r = — .009 sin a +.007 cos a. 



Harvard College. — Observations in 1871, r = — .010 sin a +.003 cos a. 



After reducing the equinox of the different catalogues to that of the 

 standard catalogue, and subtracting from the residuals given by direct 



VOL. I. 17 



