with Remarks on the Law of Frequency of Error. 41 



ing to their magnitudes. In the one case the magnitudes are 

 supposed to be wholly due to the various combinations of 17 

 alternatives, and the elements of the drawing are obtained from 

 the several terms of the expansion of (l + l) 17 , all multiplied 



into 7 . These form the following series, reckoning to the 



nearest integer; and their sum, of course, =1000 : — 0, 0, 1, 5, 

 18, 47, 95, 148, 186, 186, 148, 95, 47, 18, 5, 1, 0, 0. In the 

 figure these proportions are protracted so far as possible ; but 

 the numbers even in the fourth grade are barely capable of being 

 represented on its small scale ; after the fourth, the several grades 

 are manifest until we reach the corresponding point at the oppo- 

 site end of the series. Then, with a free hand, a curve is drawn 

 through them, which gives as their mean value 8*5, as it ought 

 to do. Now, referring to our p and q at the 250th division from 

 either end, I measure the value of q — m (or m—p), which is the 

 unit to which I must reduce any other ogive that I may desire to 

 compare with the present one. Also I can find the values 

 for m + 2(q — m) and m + S(q— m), which is going as far as a 

 figure on this small scale admits. I now protract the central 

 portion of an exponential ogive to the same scale, horizontally 

 and vertically. Not knowing its base, I start from its middle 

 point, placing it arbitrarily at a convenient position in the pro- 

 longation of the m of the binomial ; and I lay off, in the prolon- 

 gation of p and ^points that are respectively 1 unit of probable 

 error less and greater than m. The Tables of the law of error 

 tell me where to lay off the other points ; and so the curve is 

 determined. It must be clearly understood that whereas in the 

 figure both the ogive and the base are given for the binomial 

 series of 17 elements, it is only the ogive that is given for the 

 exponential, there being no data to determine the position of its 

 base. The comparison is simply between the middle portions 

 of the ogives. To speak correctly, I have not actually used 

 the exponential Tables to draw the exponential curve, but have 

 used Quetelet's expansion of a binomial of 999 elements, the 

 results of which are identical, as he has shown, with those of 

 the exponential to within extremely minute fractions, utterly 

 insensible in a scale more than a hundred times as great as the 

 present one. 



I find the position of the various points in the two ogives, 

 measured from the appropriate end of the base, to be as is ex- 

 pressed in the following Table ; — 



