DIY History | Transcribe | Scholarship at Iowa | Theory of least squares applied to the problems arising in our observatory by Arthur George Smith, 1895 | Theory of Least Squares Applied to the Problems Arising in our Observatory by Arthur George Smith, 1895, Page 26

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Theory of least squares applied to the problems arising in our observatory by Arthur George Smith, 1895

Theory of Least Squares Applied to the Problems Arising in our Observatory by Arthur George Smith, 1895, Page 26

[[page#]]22[[/page#]]

[[3 columns listed in reading order]]

==========================
||      t         | [delta]  | [delta delta] ||
||-------------|-----------|-----------------||
||   19.28    |   -.57    |    .3249       ||
||   18.38    |  +.33    |    .1089       ||
||   19.43    |   -.72    |    .5184       ||
||   19.11    |   -.40    |    .16           ||
||   17.89    |  +.82    |    .6724       ||
||   18.45    |  +.26    |    .0676       ||
||   19.50    |   -.79    |    .6241       ||
||   18.57    |  +.14    |    .0196       ||
||   18.56    |  +.15    |    .0225       ||
||   18.99    |   -.28    |    .0784       ||
||   18.52    |  +.19    |    .0361       ||
||   18.84    |   -.13    |    .0169       ||
||   18.90    |   -.19    |    .0361       ||
||   18.80    |   -.09    |    .0081       ||
||   18.76    |   -.05    |     0025       ||
||   18.55    |  +.16    |    .0256       ||
||   18.76    |    .05    |    .0025       ||
||   18.79    |    .08    |    .0064       ||
---------------|----------------------------||
                  | [delta delta] 2.7310  ||
                  -----------------------------

Set E1 = mean error of single arithmetical mean or t.
R1 = probable error of t.
E0 = mean error of weighted mean.
R0 = probable error of the weighted mean.
Then,
E1 = squareroot of ( [delta delta] / (m-1) ) = squareroot of (2.731 / 17) = +- 0.41
R1 = squareroot of ( [delta delta] / (m-1) ) x 0.6745              = +- 0.276
E0 = squareroot of ( [delta delta] / [m (m-1)] ) = squareroot of (2.731 / 17.18) = +- 0.094
R0 = squareroot of ( [delta delta] / [m (m-1)] ) x 0.6745                   = +- 0.065

This gives [[?for or per?]] the value of the interval of the transit reticule (18.71 +- 0.063) / 100 of a revolution of the azimuth screw.

Since errors in linear measurements vary as the square root of the length error for one revolution is [[? t1 ?]] error of 18.71 / 100 revolution as (squareroot of 100) : (squareroot of 18.71)

Error one revolution is [[? t1 ?]] 0.063 :: (squareroot of 100) : (squareroot of 18.71) = 0.145

Therefore one revolution = 5.345 +- 0.008 interval of transit reticule.

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[[page#]]22[[/page#]] [[3 columns listed in reading order]] ========================== || t | [delta] | [delta delta] || ||-------------|-----------|-----------------|| || 19.28 | -.57 | .3249 || || 18.38 | +.33 | .1089 || || 19.43 | -.72 | .5184 || || 19.11 | -.40 | .16 || || 17.89 | +.82 | .6724 || || 18.45 | +.26 | .0676 || || 19.50 | -.79 | .6241 || || 18.57 | +.14 | .0196 || || 18.56 | +.15 | .0225 || || 18.99 | -.28 | .0784 || || 18.52 | +.19 | .0361 || || 18.84 | -.13 | .0169 || || 18.90 | -.19 | .0361 || || 18.80 | -.09 | .0081 || || 18.76 | -.05 | 0025 || || 18.55 | +.16 | .0256 || || 18.76 | .05 | .0025 || || 18.79 | .08 | .0064 || ---------------|----------------------------|| | [delta delta] 2.7310 || ----------------------------- Set E1 = mean error of single arithmetical mean or t. R1 = probable error of t. E0 = mean error of weighted mean. R0 = probable error of the weighted mean. Then, E1 = squareroot of ( [delta delta] / (m-1) ) = squareroot of (2.731 / 17) = +- 0.41 R1 = squareroot of ( [delta delta] / (m-1) ) x 0.6745 = +- 0.276 E0 = squareroot of ( [delta delta] / [m (m-1)] ) = squareroot of (2.731 / 17.18) = +- 0.094 R0 = squareroot of ( [delta delta] / [m (m-1)] ) x 0.6745 = +- 0.065 This gives [[?for or per?]] the value of the interval of the transit reticule (18.71 +- 0.063) / 100 of a revolution of the azimuth screw. Since errors in linear measurements vary as the square root of the length error for one revolution is [[? t1 ?]] error of 18.71 / 100 revolution as (squareroot of 100) : (squareroot of 18.71) Error one revolution is [[? t1 ?]] 0.063 :: (squareroot of 100) : (squareroot of 18.71) = 0.145 Therefore one revolution = 5.345 +- 0.008 interval of transit reticule.