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Dr. C. V. Burton on a Physical Basis 



that the resultant curve rapidly approaches the Laplacian 

 form, the discontinuities being continually of a remoter order. 

 Thus with two errors (fig. 8) the curve meets the axis of 

 errors and has only discontinuities of direction ; with three 

 errors there are discontinuities of curvature, and so on. The 

 curve produced by combining n errors of this type will have 

 contacts of the (n — 2)th order with the axis of errors ; and 

 the resultant of two curves with contacts of the mth and nth. 

 orders respectively has contacts of the (w + ?n + 2)th order; 

 for contacts of the mth and nth orders would be produced by 

 combining m + 2 and n + 2 errors respectively. 



To illustrate these principles, consider the error introduced 

 into the indications of a balance by the friction of its knife- 

 edges; and to simplify the case, suppose that the balance is 

 allowed to come to rest before a reading is taken. We are 

 only concerned here with errors in the actual position of the 

 pointer, supposing the weights correct and the balance just. 

 The friction at each knife-edge is a source of error, and the 

 three sources of error are (at least very approximately) 

 independent. Let the frictional coefficient at each knife- 

 edge be constant, the maximum error due to the central 

 knife-edge being double that due to either of the others ; 

 there are then three curves of errors of the type of fig. 1. 

 The two equal sources of error being compounded will give 

 the curve ALB (fig. 10), whose maximum error is equal to 



the maximum value of the remaining error. We shall thus 

 have two equal masses, one moving rapidly backwards and 

 forwards along C D with numerically constant velocity ; the 

 other moving slowly from A to B with a slowness proportional 

 to the corresponding ordinate; and the expression (1) will 

 evidently be applicable. Consider an element hoc at H; 

 then the second factor of (1) = chance that the slowly- 

 moving mass lies between P and Q = the area PRLB. As 

 LP moves from E to F, P Q moves from the position in 

 which Q coincides with A, until P coincides with B, The 

 resultant curve of error will consist of three arcs of equal 

 parabolas (fig. 11) with foci at P, Q, and E. For deter- 



