386 



THE SAY RIO OBSEEVATOEY. 



As appears from the example, one morning observation may be combined with the 

 mean of two or more evening observations or vice versa. Also the same morning or 

 evening observation may enter into more than one combination ; thus the morning 

 series above is combined with each of the two evening series. 



p Y and j» 2 being the weights of the two equations, that of the difference equation is 



Pi fP» 



When the same observation equation enters into two combinations it is given in each 

 the weight 0.7p. 



When two equations are combined into a single mean it is given weight 2p. 



When three or more equations are so combined the weight is 3p. 



In this manner were formed 1219 difference equations. When these resulted from 

 observations made within an interval of a few days, their coefficients were so nearly equal 

 that frequently it was advantageous to combine several difference equations into one. 

 This was done by the simple process of addition, as shown by the following example : 



— 1.09a -|- 1.05?/ | 1.15* + 1.88m — 1.48s' + OMw = + 0.27 i/p = 0.64 



— 1.17 1-1.00 +1.10 1-1.43 —1.40 +0.04 —0.19 0.04 



— 1.20 +1.58 +1.00 +1.40 —1.39 +0.63 +0.32 0.64 



— 1.05 +1.06 +1.16 +1.80 —1.26 +0.66 +0.15 0.76 



— 1.11 + 1.63 + 1.12 + 1.35 — 1.25 + 0.05 + 0.34 0.64 



2 — 5.62j; + 8.12?/ + 5.59:3 + 6.89m — 6.72b' + 3.23w — + 0.89 ^p 0.305 

 — 1.71* + 2.48?y + 1.70s -| 2.10m — 2.05s' + 0.99w = + 0.27 



The final equation is the summation equation multiplied by the square root of its 

 weight. The latter is derived as follows: 



Let pi . p-2 ■ ■ ■ ■ Pn be the weights of the individual equations, the weight of the 

 sum is 



(P\ -V-L P^ 



(Pi- Vt- ■ ■ -P») I (P< ■- 



,Pt)+ ■ ■ ■ + (Pi ■ ■ ■ ■ P*~l ■ Pn) 



By tabulating the terms of this formula for the values of p which actually occurred 

 the weights were readily computed. 



By this process of combining, the number of equations was reduced to 190. These 

 were then combined in the usual manner to form the six normal equations which follow : 



697.1376a — 82.5669?/ | 239.3309a — 019.6463m + 74.3262s' — 198.3408m = — 32.0961 



+ 664.8606?/ + 544.5932s + 408.5788m — 99.8518s' — 34.9435w = — 76.1616 



| 039.9110z + 53.0980m — 32.9367s' — 68.8908m = — 84.4461 



+ 757.0544m — 112.3893b' + 107.7800w = — 12.3723 



■ | - 426.7316s' -|- 1.4786m) b= — 35.2360 



+ 126.1514w = + 13.5975 



