144 



PROFESSOR TAIT ON THE LAW OF FREQUENCY OF ERROR. 



double curvature of which this circle is the projection. But, besides being 

 foreign to our subject, these theorems follow at once from well-known properties 



12. Returning to equation (12), it is obvious that a and M must be connected 

 since we have to satisfy the condition that the probability that the error lies 

 between infinite positive and negative limits is certainty. Hence, as we may 



'••••. (14) 



for the chance that the error lies between x and x + 8x- we must have 



+ 00 



«/v 



-Mz=, 



ax - L • (15), 



But we know that 





00 



-J- 00 



which reduces (15) at once to the form 



M- 1 • • • (16), 



the required relation. 



13. It is obvious from (12) that large errors hare less probability when H is 

 large ; that is when h is small, if we put 



M 4 



Henee h becomes an indication of the comparative accuracy of the process whose 



zz::zT ms ' and " is thus desirawe to retain jt in *• «££z 



By (16) we have 



and therefore, by (14), we obtain 



h\/ 



7T 



1 ~- 



Ox 



/i\Ar 



foi he hance that the error lies between • and m+tm , the usual expression 



14. It only remains that we give an idea of the accuracy with which this law 

 of error is approximated to, in cases such as we have assumed as the ba si of 

 our reasoning even in a very small number of trials. For this purpose we ke 

 the case of 20 'tosses of a coin. Here the most probable result is, of comse 10 



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