for the Theory of Errors. 



483 



measured ; it will further be influenced by the moment of 

 inertia of the needle. 



Fig. 6. 



6. Enough has now been said to show that the law of error 

 to be adopted depends in some measure on the nature of each 

 special case ; we may next consider how to find the law of 

 error when two or more fallible elements are combined. To 

 commence with, take two elements whose curves of error are 

 of the type of fig. 1, the limits of error being +111-^ and + m 2 

 respectively. In the rectangle A C D B (fig. 7), let A = B 



Kg. 7. 



R 



m 9 QL 



= CM = MD = the unit of length; and let a particle be 

 chosen whose mass is numerically equal to m x . If this particle 

 be placed at P to represent an actual error = m x P, it is 

 evident that the uniform motion of the particle from A to B 

 represents the distribution of errors between the limits 7%OA 

 and WiOB, that is, between +m lm The second source of error 

 may be similarly represented by a particle of mass m 2 which 

 moves uniformly from C to D. When m x is at P and m 2 at Q, 

 the resultant error w T ill be mfiT -f wi- 2 MQ = {mi + m. 2 )^S, 

 where S is the centre of mass of m x and m 2 . 



To form the most general series of combinations, let m 2 

 move backwards and forwards very rapidly between C and D, 

 always with (numerically) the same velocity, while at the 

 same time m 1 moves uniformly and very slowly from A to B. 

 By following the movements of the mass-centre along E F, 

 we shall find the law of frequency of the resultant error. 

 Join PC, P D, cutting E F in Gr and H ; then while m 1 is 

 passing through P, the mass-centre is moving uniformly 

 between G and H ; and as m x moves from A to B, GH moves 



