OF ARTS AND SCIENCES. 171 



separate fundamental stars, observed January 5, 1872. They are 

 given on page xxiii., Vol. X., of the Annals of the Observatory. Here 

 a: is a complex function. It involves not only the error of reading the 

 microscopes, but several other classes of errors. They may be enu- 

 merated as follows : — 



Errors depend- r ( a ) Error of reading microscopes, 

 ing on the -j 

 observer. I (^) Error of bisection of the star observed, or its equivalent. 



' (c) Errors of graduation of the circle, both accidental and sys- 

 tematic. 

 Errors depend- (d) Error depending on the micrometer screws of microscopes, 

 ing on the , (e) Error due to the flexure of the instrument, 

 instrument. (/) Error due to an imperfect figure of the pivots. 



(g) Error resulting from a change in the position of the in- 

 strument during the observations. 



Errors inde- 

 pendent of 



(h) Error resulting from an erroneous place of the funda- 

 mental star observed. 



the observer -| (t) Error depending on the state of the atmosphere, including 

 and of the the constant of refraction, imperfect thermometers, ba- 



instrument. [ rometers, &c. 



In this case, then, one must place quite a different interpretation 

 upon the probable error of the mean value. In fact, the only safe 

 interpretation that can be given to it, is the one which regards it as a 

 means of comparing observations made by different observers under 

 nearly the same conditions and in the same manner. 



This subject may be considered in another way. It is a property 

 of the arithmetical mean that it makes the sum of the squares of the 

 residuals a minimum. The solution of a greater number of equations 

 than the unknown quantities which they contain, by the process of 

 least squares, rests upon the same basis ; viz. that such values must 

 be given to the unknown quantities as will, when substituted in the 

 original equations, make the sum of the squares of the residuals a 

 minimum. Theoretically, any unknown quantity may be made equal 

 to a constant plus the sum of all the corrections which make up this 

 quantity. We may always have 



X= C + Aa + Bb+Cc + Dd, &c. 



The only limit to the number of terms is the one which requires 

 that the coefficients A, B, C, I), &c. shall be known. The solution of 

 a series of equations of this form will give the most probable values of 

 the constant C, and of the unknown quantities a, b, c, d, &c, provided 



