138 ESSAY ON PROBABILITIES. 



8584-9, 39601, 94249, 66564, 37249, 484, the sum 

 of which is 605831; and twice this is 1211662. 

 Divide 225, the square of 15, by 1211662, which gives 

 0001856953, the weight of the average. The square 

 root of the weight is -013627, which multiplied by 50 

 gives *6814, the value of t: that of H is then -66. 

 So that it is more than 3 to 2 that the true result of 

 the preceding very discordant observations lies between 

 1001 and 1101 



To use the second table, multiply -013627 by 130 

 which gives 1 '771 51, by which divide 62, giving 35 

 very nearly. This is the probable error, so that it is 

 an even chance the result lies between 1051 35 and 

 1051 + 35, or 1016 and 1086. Divide 50 by 35 

 giving 1-43 the value of t ; for which in table II., K 

 is -66, as before. 



I now proceed to explain the meaning of the term 

 weight, as used above. When an observer has made 

 various observations, one or more of which he thinks 

 superior to the rest, as to the favourableness of the 

 circumstances under which they were made, it follows 

 that the good observations should tell more in the 

 formation of the most probable result than the indif- 

 ferent ones. If for example, a remarkably good trial 

 give 10 and an indifferent one 11, it is not reasonable 

 to say that 10^ is the most probable result. If the 

 first observation be remarkably good, it may seem not 

 unfair to give it the force of four observations, or to 

 let the number 10 have the weight which would result 

 from four observations giving 10, 10, 10, 10, a fifth 

 giving 11. On this supposition the average is the 

 fifth part of 51, or 10, instead of 10 J. This was called 

 giving the observations 10 and 11 weights of 4 and 1, 

 and the method of finding an average is this : multiply 

 every observation by its weight and divide the sum of 

 the products by the sum of the weights. Such a method 

 was adopted before the theory of probabilities was ap- 

 plied to the subject, as a direction of common sense. 

 When that theory came into use, it was found that the 



