in the Position of a Point in Space. 129 



The value of z becomes zero when we take either x={) or y=0, or 

 when xz=cti or y=oD . Hence, to determine C, since the sum of all 

 the values of z is unity, we have 



dxd>^J, y„ ^<^^<^y=h (26) 



which is equivalent to 



C^{dxdy) r^ ,a;^&.-i -a,x^, . 

 Ti -27 / V^^^) e diax) 



or to 



When the value of C obtained from this is substituted in (25), we 

 get 



This is the approximate equation of the limiting surface. It will 

 be most convenient if we restore the origin to the point where it was 

 at first, the center of gravity of all the masses s, or in other words, 

 the arithmetical mean of all the points of error, each taken with a 

 weight proportional to the probability of its occurrence. Comparing 

 (22) and (23), it appears that the origin was removed from this point 

 by substituting 



./■— ttjjj and y—^^'2p2 



in place of a; and y, so that to restore it, we substitute 



.'• + «i^j and y + «2^2 ('^'^) 



for .1' and y in (29). Employing also the known formula 



yT;0 = (-^yVf-Ul+— + -^ - etc.), (31) 



\e/ r \/i/\ 12h 288^/2 ^' V 1 



with Kj and Kg as auxiliary letters 

 1 1 



K.,=:H irr-\ -. — -„^-T-o — etc., 



12a,'Z»^ 288(a,'6,)' 



1 1 , s 



K„= 1 H 5-r + T-iT\i — etc., (32) 



12rt/6, 288(a„'^'„)' ' ^ ^ 



we find that (29) reduces to 



