190 On a New Method of Reducing Observations. 



think, in the case of many unknown variables. At the same 

 time that labour is more " skilled." There may be needed the 

 attention of a mathematician ; and, in the case of many un- 

 knowns, some power of hypergeometrical conception. Perhaps 

 the balance of advantage might be affected by an a priori 

 knowledge of an approximate solution. Which is, I think, a 

 greater convenience in the case of the new method than of 

 the received one, where the given equations are linear. Again, 

 in cases where Discordance or other irregularity may be sus- 

 pected, it will be useful to verify the result of the Method of 

 Least Squares by the simpler method. The consilience be- 

 tween the two results which Mr. Turner's work exhibits is 

 surely very satisfactory. 



Moreover, in cases where the observation-equations are of 

 the second (or higher) degrees, it seems possible that the 

 Median should have a decided advantage over the received 

 method. According to the latter, we must start with an 

 approximation sufficiently close to warrant the neglect of 

 those terms of the equations which involve the variables in 

 the second (or higher) degrees. Otherwise the principle of 

 Least Squares would lead at best* to cumbrous simultaneous 

 equations of the third (or higher) degrees. But, according to 

 the principle herein set forth, a solution is obtained by taking 

 x and y such that R, the sum of the residuals of the type 

 \ax 2 + 2hxy-\- by 2 + 2fx + 2gy — r~\, each taken positively, should 

 be the least possible. By considering the geometrical inter- 

 pretation of this condition, we may see that the required point 

 is on an observation-curve. As in the simpler case, we may 

 climb to the position of highest probability by noting the value 



of -j- at every turn. Thus, let us start from a point x , y 



on the curve a x 2 + 2h o xy + b y 2 -\-2f o x + 2g y = r o : and, as 

 before, let the coefficients of the observation-lines below and 

 left of the initial point be dotted ; above and right, plain. Then 



a r R=2dx[{Sa / -Sa)x + (S//-S% + (S/'-S/)] 



-r2<fy[(S&'-S% + (S//-SAK+ (S/-fy)] ; 

 where 



dx[a x + h y +/ ] + dy [b y + h x + go] = 0. 



rITt 



From these equations -p is to be found. If it is positive, we 

 move backwards, and vice versa. At each intersection we 

 take the path for which -j- is least. 



* Even supposing the law of facility to be the Probability-curve. 



