A0~ ESSAY ON PROBABILITIES. 



to be 5 divided by twice E 2 , the preceding methods 

 will show the chances of a large number of observ- 

 ations, upon a supposition intermediate between the 

 two last, and coinciding nearly with the first when the 

 errors are small, and with the second when they are 

 considerable. 



I now proceed to the method of combining the 

 results of observation, and deducing the mean risks of 

 error. Suppose, for example, that A and B are two 

 results of a large number of observations, of which the 

 product is required. Nothing can be more erroneous 

 than to suppose that the mean risk of this product 

 will be the product of the mean risks of its factors. 

 By the mean risk here is meant the same thing as 

 before : imagine the product of A X B formed in every 

 possible way from the single results of the several 

 observations. Each error will be as likely to be positive 

 as negative, if the errors of the original observations 

 be the same. Take the average of all the several 

 errors, neglecting their signs, and one half of this 

 average will be (in the long run) what is called 

 the mean risk, explained in the same manner as in 

 p. 135. 



The risk of the result will be modified by the 

 manner in which the operation makes one quantity 

 affect the error of the other. Suppose, for example, 

 that observation gives A 100, and B = 150. If 

 these were certainly true, the required result would be 

 accurately 100 x 150, or 15,000. But if 100 be 

 wrong to any amount, the product will be wrong by 

 150 times that amount, on that account only : while, 

 if B be wrong, its error will be multiplied 100 times 

 in the result ; besides which, there will be an additional 

 error, the product of the two errors of A and B. On 

 the other hand, if observation give A too large and B 

 too small, the opposite errors may either compensate 

 each other exactly, and give the product precisely what 

 it ought to be, or may make some approach towards this 

 compensation. The product then may be rendered 



