154 ESSAY ON PROBABILITIES. 



only to be trusted when the mean risks are small. 

 Let A= 100, B = 150, and let the mean risks of A 

 and B be 1 and 2. Then 1 is -01 of 100, and 2 is 

 0133 of 150. The squares of -01 and -0133 are -0001 

 and -00017689, the sum of which is -00027689, the 

 square root of which is -Ol6'7. This multiplied by 

 100 x 150, or 15,000, gives 250-5, which is the 

 mean risk of the product 15,000. 



4. To find the mean risk of a fraction, or of the 

 quotient of a division, multiply each term (numerator 

 and denominator, or dividend and divisor) by the mean 

 risk of the other, add the squares of these products and 

 extract the square root of the sum: divide this by the 

 square of the denominator or divisor ; the result is 

 the mean risk required. But if the fraction be very 

 small, it is sufficient to divide the mean risk of the 

 numerator by the denominator ; while if the fraction 

 be very great, it is sufficient to multiply the fraction by 

 the risk of the denominator, and to divide the result by 

 the denominator. 



The preceding will serve as specimens of the manner 

 in which complicated results of operation can have 

 those probabilities investigated which depend upon the 

 probabilities of error in their constituent parts. It 

 would be impossible to lay before a reader unacquainted 

 with the differential calculus, any such digest of rules 

 as would enable him to treat all cases with facility. 

 Any one of the mean risks obtained above will serve to 

 determine, as in p. 139-, the weight of the result, from 

 which its law of error may be investigated, as in 

 p. 143. 



It appears that the chances of error may be con- 

 siderably multiplied in the course of the operations to 

 which the results of observation are subjected. It 

 must, therefore, be the object of an inquirer not only 

 to make good observations, but also to select such 

 methods of observing, and such methods of treating 

 the observations (the latter generally depending upon 

 the former), as will render the final error the least 



