§ l6 OF ARTS AND SCIENCES. 417 



cial case, that the velocity or error is positive and finite, 

 W'e are able, nevertheless, to calculate from various data 

 the average, the mean, and the probable value. The 

 chances are the same for positi\'e as for negative magni- 

 tudes, and the principle of the conservation of energy 

 requires that velocities taken at random, like errors, shall be 

 compounded so that the mean square of the resultant may 

 be equal to the sum of the mean squares of the com- 

 ponents. 



If we possessed absolutely no know^lege of the mechan- 

 ism by which velocities are determined, our only choice 

 would be to apply the same rules as to accidental errors; 

 and by comparing the ditferent formulae which have been 

 suggested for the kinetic theory, one might easily become 

 convinced that the substitution of one formula for another 

 is not likely to cause a mistake of more than one place in 

 the decimal point. 



The formula for the probability (^c) of a velocit}^ of a 

 gas molecule^, for instance, being less than r, may be 

 deduced from Watson's formula, in his Kinetic Theory 

 of Gases, page 5, namely. 



V 77 ^ ° . 



while Chauvenet's formula for the probability that an error 

 will be less than /is 



4>t = -^ r Vv//. 



The formulae for solids and liquids have not apparently 

 been worked out, and, in complete ignorance of the mech- 

 anism which determines the kinetic energy of a given 

 particle, we can obtain an approximation, only, to the dis- 

 tribution of velocities, b}' means of the more general theory 

 of probability. For convenience of reference, a table of 

 the probability of errors has been appended, calculated by 



