for the Theory of Errors, 485 



lowing the method of the previous section, let the mass m 1 

 move very rapidly backwards and forwards between A and U 

 or between A and B, having at each point of its path a velocity 

 inversely proportional to the corresponding ordinate ; (adding 

 U B to the path makes no difference, since the velocity in this 

 part would be infinite) . Similarly let m 2 move from C to D 

 with a very small velocity, which follows a similar rule. First 

 of all let the mass m 2 be passing through R (NR = # 2 )> so that 

 the second error has the value m 2 x 2 ; also in fig. 9 let MT = ^ 1? 

 TT = Sx 1 , 0$ = a/, SS' = &i/. Then the chance of a resultant 

 error between (m 1 + m 2 ) x' and (rrii + m^) (x r + haf) — the chance 

 that m 1 is moving between T and T' 



=y x hx x -T- area ALU. 



m 1 + W? 2 * / AT TT 



= Vi— -6x f -r- area ALU. 



m 2 



For simplicity, let the scale of ordinates be chosen so that the 

 area ALU = VKD = unity ; then the chance that R, may lie 

 between x 2 and x 2 + dx 2 =y 2 dx 2 . 



Hence, taking both movements into account, the chance of 

 an error between (m x + m 2 )x / and {m 1 + m 2 ) [x' + dx f ) 



m x + m 2 ,r 

 = —^—dx)y 1 y 2 dx 2 ; 



y x and y 2 being connected by the relation 



m 1 x 1 + m 2 x 2 = (m, + m 2 )x r , 

 and the limits of integration being determined by 



With similar notation for n independent sources of error, 

 we shall have for the chance of a resultant error between 

 (m x + . . . + m n ) x' and {in Y + . . . + m J (a? + dx') 



, , (m y + m 2 ) (mi + m 2 + m 3 ) . . . (m 1 + m 2 + . . . + m n ) 

 m 2 m z . . . m n 

 X jj. • . j>iy 2 y3 • • • y n dx 2 dx z . . . dx n ; . . (2) 



where m x x x + m 2 x 2 + . . . + m n x n = (m^ + m 2 + . . . + m^)x f , 

 and the limits of integration are given by 



a?i*<l, x 2 2 <l,...x n 2 <l. 



If we make n infinite, each of the y's equal to J, and all the 

 m's equal, this reduces io the case from whose approximate 

 solution Laplace deduced his law of error. 



8. If we follow up the operation of § 6 by successively 

 adding sources of error of the type of fig. 1, we shall find 



