414 Historical Note on the Method of Least Squares. 



by M, the ordinate of a curve to the abscissa x, we shall have 



u = e " , which is the general equation of the curve of 



prohahility. 



"When only the maximum of probability is required, we 

 have no need of the values of e, a' and m; it is proper, how- 

 ever, to observe that m must be negative. This is easily shown. 

 The probability that the errors x, y, z, etc., happen in the dis- 



which is equal to e ^ 2a 26 2c / ^ ^^^ ^^^^ 



quantity will evidently be a maximum or minimum as its index 

 or logarithm is a maximum or minimum ; that is, when 



Now when x + y+z, etc., = E, we know that 



-— +-T-H , etc., = mmmium, when —=^=- 



and therefore - ] ^+^+-^, etc., \ = maximum. 



IV 



When ^==1=' 



, it is evident therefore that m ] 



negative ; and as we may for the case of maxima use any value 

 of it we please, we may put m = -2, and the probability of x 



in a is w = e^ «^. If we put |^ = -1 and a'=f, we have 

 " = e(^*~^') for the equation of the curve of probability ; but if 

 we suppose /=^ = 0, the ordinates u will still be proportional to 

 their former vabies, and we shall have u = e"^' or w = — , which 

 is the simplest form of the equation expressing the nature ot 

 the curve of probability." 



Immediately following the above general solution by Dj"- 

 Adrian there are given applications of this method to the fol- 

 lowing problems. 



1. To find the most probable value of any quantity of winch 

 a number of di 



2. To find a most probable position of a point in space. 



3. To correct in the most probable manner the dead reckon 

 Qg at sea. 



4. To correct the bearing and distances of a field survey. 



