OF ARTS AND SCIENCES. 610 



ephcmevis given by Dr. Sclionfeld * for the present year differs by 



thirty-five minutes from his formuhi, or agrees within two minutes 



with the result of tlie present observations. The writer has shown 



in anotlier placet t'l'i't observations show a deviation from Schonfeld's 



formula of twenty-nine minutes at the eiid of 1878, and that this 



deviation is increasing at the rate of about three minutes a year, 



which would also give a correction of thirty-five minutes. 



Any portion of the observations, as those of a single observer, or of 



one evening, would in general be better satisfied by moving the curve 



horizontally, or by assuming a different time of minimum. We wish, 



therefore, to know what correction t to the minimum is indicated by 



such observations. Let H equal the residual found by subtracting 



the value given by the assumed curve fi'om that found by observation, 



and let r equal the residual when the minimum is altered by t. Also, 



let a equal the difTerential coefTicient of the light in terms of the 



time, or the cliange of the light in magnitudes per minute. Then 



Ji = r + at, in which r and t are unknown. Solving with regard to t, 



li r 



we obtain, i = . The weight to be assigned to such a deter- 



a a ® ° 



mination of t will be proportional to a, since the errors are almost 

 entirely due to erroneous determinations of the light, the error iu 

 the time being wholly insensible. Accordingly the effect on t of an 

 error of a hundredth of a magnitude will be inversely as the rate of 

 change of the light, or the weight should be proportional to a. What- 

 ever the sign of a, the weight must always be positive. Multiplying 

 the above value of t by a, we have at =^ ±, R -^z r, in which a is 



R r 

 positive, and the signs of R and r will always be those of — and . 



Taking the sum of all these equations, we obtain 2a< =: Ai? — Ar, in 

 which 2 denotes the arithmetical sum of all the separate values, 

 A their algebraic sum, taking into account the signs assigned them 

 above. But 2a< = iSa, since, although t is unknown, it is the same 

 for all the observations. Again, Ar = 0, since the separate values 

 of r are arranged according to accident. Therefore, t%a = AR, or 



t =■- - . The computation is made by taking the algebraic sum of 



all values of R after changing the signs of those, in which a is nega- 

 tive, and dividing the result by the arithmetical sum of all the values 

 of a. 



Vierteljalirsschrift, xv. 14. t Proceedings American Academy, xvi. 36. 



