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VII. Reduction of Error by Linear Compounding . 
By W. F. Sheppard, Sc.D., LL.M. 
Communicated by E. T. Whittaker, F.R.S. 
Received April 23,—Read June 24, 1920. 
Contents. 
1. Introductory 
Page 
200 
Conjugate Sets. 
2. Conjugate set.201 
3. Relations between original set and conjugate set.203 
4. Two related pairs of conjugate sets.204 
5. Sums as conjugates of differences.205 
Reduction of Error (General). 
6. General theorems.208 
7. Notation : and particular values.210 
8. Improved values in terms of conjugates.211 
9. Mean products of error of improved values.212 
10. Expression in terms of a related set.214 
11. Special case of differences.215 
Relation of “ Reduction of Error ” to “ Fitting ” (of a Polynomial). 
12. Standard system.215 
13. Fitting by least squares.216 
14. Fitting by moments.219 
15. Reduction of error.219 
16. Difference of the two processes.219 
Some Steps in the General Solution. 
17. Preliminary.220 
18. Formula of progression.220 
19. Mean products of error (alternative formula).221 
Application to Self-Conjugate Set. 
20. Preliminary.222 
21. Improved values.225 
22. Extent of improvement (central differences).230 
23. Smoothing.231 
VOL. CCXXI.—A 588. 2 F [Published October 22, 1920. 
