322 PROFESSOR L. BECKER ON THE DISTRIBUTION OF 



v = h t + x-y n (h n +x). I take the sum of the residuals irrespective of sign for n = 2 

 to 5. For each value of n the sum of the residuals is a minimum for x between 80 

 and 120, and the sum for all values of n together is a minimum for x = 110. The 

 sum of the residuals is 50 per cent, greater for x = 20 and x = 300. x = 500 (radius 

 of the sun) leaves inadmissible residuals. 



The observed values of FI and y n belonging to x = 110 give y = 3 '4 for n = 2, 

 3'3 for n = 3, 3'5 for n = 4, and 3'8 for n = 5. 



In the same way, if the distances obtained from Photograph No. Via. be plotted 

 as ordinates and those on Photograph V6. as abscissae, a straight line represents the 

 observations as well as any smooth curve that can be drawn, and the same remark 

 refers to the distances obtained from Photographs VI>. and VII. The sum of the 

 residuals is again small for values of x near 110, though the range of possible values 

 of a: is larger. The resulting value of y lies near 4. 



I think this is sufficient proof that the function represents very nearly the 

 observations. Let us assume it to be exactly correct. The method explains that x 

 and the n constants y are determined independently of the times of exposure from 

 the condition h + x/^ + x = h\ + xlk' n +x y n , and that there are n equations for the 

 unknown y. These n equations will be rigorously satisfied, provided the correct 

 values of F be introduced, and hence nl values F,, n or n l values of the time of 

 exposure can be determined from the equations along with y. 



I prefer to determine x and y together by the Method of Least Squares. Let 

 #o+f, 2/o +ij be the true values of x and y, and v n be the accidental error of measure- 

 ment of h n , and AF Mn the correction of an approximate value F mn , which need not 

 necessarily be the calculated value [5 (c)]. The observations must rigorously satisfy 

 the equations 



) 0, 



//mT'Vja-t-iCo-l-Cr 



or 



The sum of the squares of the left side which contains the accidental errors is a 

 minimum for the most probable values of , 77, and (nl) values AF mn . The solution 

 gives these unknowns as functions of one of the AF, or if all the AF be introduced as 

 unknowns one of them must come out indeterminate. Instead of AF, I introduce the 

 corrections of the adopted times of exposure A<. For the first five photographs , rj, 

 A,, A^j, A< 3 will be found as functions of A< 4 and A 8 , and for Photographs Vi., VI., 

 and VII., f, 17, A< 6 as functions of A< 5 and A 7 . Finally, all the time records can be 

 used in determining the corrections A.t. It will be seen that the uncertainty of the 

 exposure of Photograph VI. is not such a serious deficiency as might be expected at 

 first sight. 



