Mr. ELLIS, ON THE METHOD OP LEAST SQUARES. 209 



Let 6, 62. ..e„ be the errors which occur at the first second &c. observation ; d), e, rf e,, d)^ e, d e.^... 

 (p„e„d€„ be the probabilities of their occurrence : the form of the function d) determining the law of 

 probability of error, which, for greater generality, we suppose different at each observation. The 

 probability of the concurrence of these errors is of course 



^,e,^2e2...^„e„d6, ... de„ (l), 



and the first principles of the theory of probabilities show that the value of pdu will be obtained by 

 integrating (l), £i...e„ being subjected to the condition 2jue = u. 



Thus 



pdu = fipi 6t(p2e2 ■■■ (p^e^dei ... rfe„ (2) 



with the relation 



^1 Ci + M2e2-.. + M„e„ = M. 

 Consequently 



pdM = rfe„ /0,€i...^„_,6„_,(^„ "—dei...de,.i (3). 



Now by Fourier's theorem 



which, replacing — by a, becomes 



— / da (pe„cosa(u - 'S,fj.€)de„, 



»' 



TT 



Therefore 



M„de, 







pdu = — — - da / rfe, ... / de^ip^e, ... (b^e^cosa (u - 'Euls) (4). 



Now if u and e„ are to vary together 



du = ii^de„, and therefore 



P= — J da J det ... J rfe„0,ei ... ^„e„ cosa (?« - 2;u6) ., (5). 



And finally, 



P=-f du f da f rfei... /"** de„d),e, ...d)„e„cosa(M - 2/i£) (6). 



Now let us suppose that equal positive and negative errors are equally probable. In this case 

 <pe = (p (- e), and consequently, 



y_ J (j)€ sin afiede = 0. 



Hence (f5) will become 



•'■'=—/ du I cosauda f " ^,6iCosa^ii£,rfe, ... / " d>,£„cosa/i„e„rff„ (7). 



— <rj 



The next step is to find an approximate value of this expression. 

 When a = /^J0ecos afxede = f^^<pede = 1, 

 as the error e must have some value lying between ± eo . 



