322 J. F. EXCKE ON THE METHOD OF LEAST SQUARES. 



The last integral gives the area of the curve of probability 

 taken from the axis of the abscissas to the curve. It represents 

 the number of observations which are possible, and embrace 

 all errors. Each element of surface (p A d A compared with 

 the whole surface, shows the proportion which the number 

 of obser\'ations giving errors between A and A + d A bears to 

 the total number of observations ; or it gives the probable num- 

 ber of observations charged with these errors, the whole number 

 being = 1. 



The object of every observation is the deduction of one or 

 more quantities, by which the observed phenomenon is pro- 

 duced. In the places of the planets, for example, these magni- 

 tudes may be the elements of the paths of the planets and of 

 the earth. The manner of combining the elements so as to 

 obtain the observed value must be supposed known, if we 

 wish to determine the value of the elements from observation ; 

 therefore every observed quantity M will give an equation 



where the function y is known, and x, y, z are to be determined 

 according to their most probable values. The equality will be 

 more or less presented according to the values assumed for 

 X, y, z. If we suppose x ■= p, y ■= q, z = r, and if 



^ =f{p,q,r), 



then M — V would be'the error of the observations in case the 

 values p, q, r were the true ones. 



If several observations of the same kind have been made, in 

 which all the same elements p, q, r determine the observed 

 value, then, in similar manner, by the assumption of x-=.p, 

 y^q,z = r, the errors M' - V, M" - V", M'" - V" wiU be ob- 

 tained. By another assumption, x = /)', y = q', z = r' , substituted 

 in the same manner in all the equations, other values of V, and 

 consequently also other values of M — V will be obtained, so that 

 to every hypothesis as to the value of x, y, z, appertains a deter- 

 minate system of errors A, A', A", which depend on the hypo- 

 thesis. In order to determine from hence the most probable 

 values of x, y, z, we need two propositions from the calculus of 

 probabilities, one of which gives the probability of a connected 

 system of errors when the probability of each single one is 

 known ; the other teaches how to determine the probability 

 of the hypothesis from the probability of the system of errors 

 belonging to it. 



