Contents. xxix 



CHAPTER XIX. 



THE THEORY OF THE AVERAGE AS A MEANS OF APPROXIMATION 

 TO THE TRUTH. 



1 4. General indication of the problem : i.e. an inverse one requiring the 

 previous consideration of a direct one. 



[I. The direct problem: given the central value and law of 

 dispersion of the single errors, to determine those of the 

 averages. 620.] 



6. (i) The law of dispersion may be determinate a priori, 



7. (ii) or experimentally, by statistics. 



8. 9. Thence to determine the modulus of the error curve. 



1014. Numerical example to illustrate the nature and amount of the con 

 traction of the modulus of the average-error curve. 



15. This curve is of the same general kind as that of the single errors; 



16. Equally symmetrical, 



17. 18. And more heaped up toivards the centre. 



19, 20. Algebraic generalization of the foregoing results. 



[II. The inverse problem: given but a few of the errors to 

 determine their centre and law, and thence to draw the above 

 deductions. 2125.] 



22, 23. The actual calculations are the same as before, 



With the extra demand that we must determine how probable are the 

 results. 



25. Summary. 



[III. Consideration of the same questions as applied to certain 

 peculiar laics of error. 2637.] 



26. (i) All errors equally probable. 



27. 28. (ii) Certain peculiar laws of error. 



29, 30. Further analysis of the reasons for taking averages. 

 31 35. Illustrative examples. 



36, 37. Curves ivith double centre and absence of symmetry. 

 38, 39. Conclusion. 



