in the. Position of a Point in Space. 137 



- - = - (a/^, - 1 ) -ad.r, -^~- = -" (cCb., -I) -a.. dy 



%c z^ ^ ^ J 



aud integration gives 



^ tti^bi — l a^^i — l a^^ba — l —atX—a^y—aaZ ,^^. 



'W=:(Jx y z e . (60) 



The value of ?o becomes zero for a;=0 or y=0 or 2=0, or for x=aD 

 or 2/^oc or 2^00 . Hence, to determine C, we have 



-— ,^-- / / / wd,rdydz=zl, (61) 



m-aydzt/^ J ^ J^ 



and as in the«case of (26), this is shown to be equivalent to 



cr(a,-6,)7-(aA)r(a;63) ^^ 



ttl'-^fel a2^&2 «3^&3 777' 



a J a^ a.^ dxdyaz 

 When C as thus obtained is substituted in (60), we get 



a.axiAxdydz , .ai'^61 — 1, ^a^-b.—i .as'^s— 1 



•"=r(«>:)f(OT<*:) '"■•'■' ^"''^ *"•'* . 



—a^x—a^y—a^iZ 

 e . (63) 



Now let the origin be restored to the point where it was at first, 

 the center of gravitj' of all the masses w, that is to say, the arithmet- 

 ical mean of all the possible points of error, each taken with a weight 

 proportional to the probability of its occurrence. It appeal's from 

 (57) and (58) that the origin was removed from this point by sub- 

 stituting 



X — a^b^, y — «2*25 2 — «3*3> 



in place of x, y, z, so that we bring it back by substituting 



x + a^b^, y + a2^2^ 2 + «3*35 M 



for X, y, z in (63). Employing also the value (31) for /"(w), and 

 writing as in (32) 



and putting iv=Wdxdydz, we find that (63) reduces to 



w-— ^ (i+^r ■" (,+JL\' 



"-(2;r)}K,K,K.V(«,*A)V "A/ \ <'A' 



O^i) 



tts^ — l 



—aiX—aiy—aaZ 



(66) 



