APPENDIX B. 223 



It is important to notice that all sections parallel to the same 

 co-ordinate plane have the same probable error. 



4. The ellipses (2) when referred to their principal axes become, 

 after some arithmetical simplification, 



the major axis being inclined to the axis of x at an angle whose 

 tangent is O5014. [In the approximate case the ellipses are 



/\ L 77 



+ ( = const., and the major axis is inclined to the axis of x at 

 I 2 



an angle tan"" 1 !-.] 



5. The question may be solved in general terms by putting 

 YON = 0, XOM = $, and replacing the probable errors 1'22 and 

 1'50 by a and b respectively; then the ellipses (2) are, 



?r (^ -- y tan 0) 2 ,> 



equation (3) becomes 



a- 



y or tan 6 



or & = tan <b = 



x b' 2 + a 2 tan 2 



and (5) becomes __=__ + _^5 (9) 



tan d) e 2 



whence _ = _ (10) 



tan o a 



If c be the probable error of the projection of _p's whole motion 

 on the plane of xz, then 



c 2 = a 2 tan 2 + &, 



which is independent of the distance of p's line of motion from the 

 axis of x. Hence also 



tan d> a 2 



tan 6 b 2 



(11) 



Problem 2. An index q moves under some restraint up and down 

 a bar AQB, its mean position for any given position of the bar 



