1 873.] in Experimental Research, 5 



may have positive and negative signs, so are we now in a 

 position to verify these observations, or select those most 

 closely approximate to the truth. 



Bat there have been tabulated the values of the celebrated 

 definite integral — 



H, = -4- fe ~ t2 .dx; 



!•/>■ 



and from these tabulated values (for the extension of which 

 we are again "indebted to Prof. De Morgan) we may, with 

 much less trouble and more accuracy, arrive at the desired 

 result. The values have been arranged in two tables: the 

 columns of Table I. are headed thus : — 



t. 



H. 



A. 



A 2 . 



0*50 

 2*17 

 2*68 



0*52049 99 

 0*99785 11 

 0*99984 94 



874 38 

 9 96 



84 



8 88 



44 



5 



t represents every hundredth of a unit from o to 2. 



H represents the values of the area enclosed by an 

 asymptote, this asymptote continually approaching but 

 never reaching the abscissa, the whole of the enclosed 

 area forming one square unit. 



A represents the differences of these values. 



A 2 represents the differences of these differences. 



In Table II. are three columns only, t, A, and K, — a mo- 

 dification of H, — headed thus : — 



t. K. A. 



4*6 0*99808 40 



Let us now take an illustration. There are ten observa- 

 tions of which the arithmetic mean is — 

 a = 200*01577. 

 The sum of the squares of the ten differences between a 

 and each individual observation is — 



%e z =0*00000007. 

 Hence* the weight of a is — 



(number of observations) 100 



w = K =— - = = 715000000. 



22<e 2 0*00000014 



The largeness of this figure indicates the high degree of 

 probability that a is very near to the true value sought. 

 But the query may be put — What is the true value ? It will 

 be seen that the question does not admit of absolute answer, 

 and for the following reasons : — One of the observations 



