of the Modulus of Errors. 



503 



Such being our qucesitum, let us represent our imperfectly- 

 graduated data by the accompanying figure. The divisions 









• 











71— 1 







n x 







n_2 





n- d 



n 2 





n s 



P- 2 P-l P, P 2 



of the horizontal line correspond to the degrees employed in 

 the given statistics — e. g* two inches in the case of Dr. 

 Baxter's statistics — degrees usually, but not necessarily, equal. 

 They are here at first supposed equal, each of length 2i. The 

 symbols n Q , n v n_i, &c. represent the number of observations 

 corresponding to each degree, e. g. the number of men between 

 two heights differing by two inches. Let h , h 1} h-\ be the 

 heights of the corresponding columns, so that n = 2ixh. 

 Take a point at the middle (or, if it seem better, at 

 one of the extremities) of the base of the highest compart- 

 ment. The sum of the squares of errors measured from the 

 real centre of gravity is less than the sum of squares of errors 

 measured from any other point, in particular 0. The latter 

 quantity is less than what it becomes, if we suppose the 

 observations in each of the compartments outside the central 

 one to be disposed^ not as, doubtless in fact, under a descen- 

 ding curve, but in a rectangle as represented in the figure. 

 This quantity 



= n 1 OP 1 2 + n 2 OP 2 2 + &c, +n_ 1 OP_ 1 2 + &c, 



(where P 1? P 2 , P_i, &c. bisect the bases of the corresponding 

 compartments) + S-p 3 /i (exclusive of h ) + the contribution of 

 the central compartment. It remains to find a superior limit for 

 the last term of this expression. Assuming, as we have done all 

 along, that the curve with which we are concerned has only one 

 maximum, we may be sure that the points at which the curve 

 strikes the upright boundaries of the highest compartment are 

 higher than h v the height of that one of its immediate neigh- 

 bours which is lowest. The contents of the highest compart- 

 ment then consist of a rectangle at least as high as h ly sur- 

 mounted by a figure which terminates in a cap (much like, 

 but not in general so regular as, the vertical section of a 

 conical rifle-ball. The contribution to the errors about 

 which the solid rectangle of height h x makes is %i 3 h v The 



2N2 



