418 Mr DE morgan, ON THE THEORY OF 



in all the errors would change the side of the truth on which the most probable result lies. 

 That is, we must suppose that the coefficients of all sums of the second, fourth, &c. orders 

 vanish. This will presently be confirmed. 



To deduce a law of error from observation with theoretical strictness, we should require 

 to know, first, the truth, secondly, the individual results of an infinite number, o-, of obser- 

 vations. If rdv represent the number of the errors which lie between v and v + dv, it is 

 then required that we should find what function of f is T-=-cr. This function is the modulus 

 of facility. But it will be foreseen that a preferable plan would be to determine J,^^, the 

 average 2A;*'' power of an error, in terms of k, and then to investigate the form of (b which 



satisfies the equation T^t) . v^'^dv = A^ for all values of k, A^ being unity. 



I now ask what supposition we can admit as to the values of A^^. That Jjj should be 

 finite for all values of k is obviously indispensable : no law of error which allows large 

 errors to occur so frequently that the average tenth power, for instance, of an error, increases 

 without limit with the number of observations, is worth consideration for comparison with our 

 experience. Thus it would be absurd to contemplate any result derived from the modulus 

 1 : TT (1 + a?'), in which even the average error, independently of sign, is infinite. Further, 

 we cannot doubt that all observations are made under laws which, if the units of measurement 

 be sufficiently great, must give A^^. diminishing without limit as k increases without limit. For 

 the errors will then be always fractional parts of a unit, and A^ must diminish without limit 

 as k increases. 



The final modulus, y/c.e~"^ : -v/tt, does not satisfy this condition. Be the unit of 

 measurement what it may, y/j* increases without limit with k. The transit observer has 

 learnt to use and to be satisfied with a modulus which asserts and takes into theory the pos- 

 sibility of an error of a century at a single wire. Reckoning in seconds, let c = 25, which 

 gives a probable error of little less than 0^1 on each wire, and may nearly represent a 

 tolerable observer. When k is great, ^jt ^^ nearly -y/Z. {k : ce)*. The average 100th power 

 is about eighty-four thousand millions of millions, as great as it would have been if every 

 error had been but little less than l',5. That is, errors of more than 1^5 occur often enough 

 to compensate those which are less, in the summation of 100th powers. Now, — not speak- 

 ing of errors of clock-reading which, though errors, are self-detecting, and are corrected, and 

 are truly no more connected with the errors we are speaking of than those which arise from 

 setting the transit for a wrong star — we know that I ',5 of actual error of observation is 

 never made. The defence of our modulus lies in its sufficiency so far as As and ^4 are con- 

 cerned, beyond which we have no occasion to use it. 



Since / (p.v . oi?^dx is to be finite for all values of k, it is clear that (px must be of the 



transcendental character: and since (pai must be even, e~*^ would seem at once to be the form 

 on which we must depend. This function made its appearance as the means of expressing 

 results connected with high numbers, in the hands of De Moivre, in the second edition (1738) 



