82 Mr. Ivory on the Method of the Least Squares. 



f(e) = <p{ — e): and, as it is most reasonable to suppose that 

 f {e) is a continuous function, at least within certain limits, it 

 follows that <p{e'^) is the proper expression of the probability 

 of the error e. 



All the errors being contained between certain definite li- 

 mits, suppose +y; the function ^{e'^^) will be evanescent 

 when e is equal to or greater than +/'. Therefore, strictly 

 speaking, ^ (^) must be a discontinuous function, having real 

 values only between the limits +J', and no values at all be- 

 yond the same limits : but this condition will be sufficiently 

 fulfilled if ip (e^) be evanescent when ^ is infinitely great, and 

 have a very small value when e' =f. 



The function tp (e') may be regarded as the ordinate of a 

 curve, corresponding to the abscissae + e ; and the small area 

 de (p (^) will denote the probability of an error between the 

 limits e and e + de. Wherefore the integral J" d cf (e"^), taken 

 from e = a to e = b, will express the probability of an error 

 between the limits a and b. And because all possible errors 

 are contained between +^ or + cvd; it follows thatyc?e(p(«*), 

 between the same limits, must be equal to unit, the expression 

 of certainty. 



These are properties which it is essential that the function 

 expressing the probability of an error must possess. Let us 

 next consider the probability of the simultaneous existence of 

 a number of errors. 



2. The several errors e, e', e", &c., are independent of one 

 another, since they arise from separate observations ; their re- 

 spective probabilities are, (p {e% (p {e"^), (p (/'*), &c. ; wherefore, 

 by the known rules, the probability of their simultaneous ex- 

 istence is the product, 



<p{e''-).<p{e'^).^{e"-)hc. (P). 



Now as every factor is always positive, and is evanescent 

 when the error is equal to + oo ; it follows that the product 

 will have a maximum relatively to every error, and hkewise 

 an absolute maximum for certain definite values of all the 

 errors. If the errors be functions of .r, y, &c., the equations 

 of the several maxima will be, 



" " + &c. = 0, 



4 &c. = 0, 



and all these equations together will determine the values of 

 x,y, &c. which correspond to the absolute maximum. 



Again, let \t/ denote a rational and integral function of e*, 

 e", r', &c., consisting of positive terms only. The function 



will 



