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



added together : it is therefore proportional to the product 

 (P), or to the exponential quantity 



c = c 



"Wherefore leaving out the constant factor, the probability of 

 S . e\ or, which is the same thing, the probability of the 

 simultaneous existence of the eri'ors 7i, v, iv, is proportional to 



c 

 In order to find the separate probability of the error m, we 

 must take the sum of all the values of the tbregoing expression 

 that arise by combining u with every possible value of v and 

 w. it is therefore proportional to the fluent, 



J dvdwc , 

 both the integrations being executed between the limit +c\3. 



To determine the integral, expand the squares in the value 

 of A, and collect the like terms ; then 



\ = ti^ S . a^ + v^ S .b'+ 11^ S .c^ 



+ 2uvS .ab + 2uwS.ac + 2 vwS .be. 



Again, assume i = Pm, 



f = P'u + QVy 



r=P"«H-Q'T>+ Rw, 

 and determine the arbitrary coefficients so as to satisfy the 

 condition 



\ = t^ + t'^ + ^'K 

 By equating the coefficients of the like terms of this equation, 

 we shall get 



px + p'. + p''» = S . a', P'Q + P"Q'= S . a 6, 



Q'+Q'»=S.JS P"R = S. ac, 



R^ -S.c\ Q'R = S.bc. 



Hence we obtain P* = — ; the values of M and N being as 



follows, viz. 



M ~S. a^ xS.b' xS.c^ + 2S. ab xS.ac xS. be 



-S.a*x{S.bcY-S.b^x{S.acY-S.c^x{S.a b)*. 



N = S.b'xS.c'—{S.bcY: 

 and it is easy to prove that M and N will be always positive. 

 It will not be necessary to determine the other coefficients. 

 We shall now have 



J dvdwc =-^r^.Jdtdtc , 



the limits of the integrations being the same as before. 

 Wherefore the probability of the error u is proportional to 



-£-''\rdt'c-''"*'\fdt"c-''''""; and 



