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, 



+ -d. — = constant, . . . . . (6) 



20-68 5-92 ' v 



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

 tangent is - 5014. [In the approximate case the ellipses are 



/2 2 



— — + — — = const., and the maior axis is inclined to the axis of x at 



7 2 J 



an angle tan -1 -!-.] 



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

 YON = 0, XOM = <f>, and replacing the probable errors P22 and 

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



?y 2 , (« - V tan Of _ fi n 



~,+ p C < ...... (7) 



equation (3) becomes 



^~ + tan ^ = 



a 1 tr 



x ^ b 2 + a 2 tan 2 # 



y , . a 2 tan 



or - = tan <h = 



and (5) becomes __ =_ + __ — ....... (9) 



e'- 4 a 2 b- 



whence ? =— , (10) 



tan b 2 v ' 



If c be the probable error of the projection of ps whole motion 

 on the plane of xz, then 



c 2 = a 2 tan 2 + b 2 , 



which is independent of the distance of ]/s line of motion from the 



axis of x. Hence also 



tan cf> _a 2 ,, , s 



ta!T^ _ ~P • • • ( ) 



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 



