APPENDIX B. 223
It is important to notice that all sections parallel to the same
coordinate plane have the same probable error.
4. The ellipses (2) when referred to their principal axes become,
after some arithmetical simplification,
x2 y2
20 68+ 592 = constant,
the major axis being inclined to the axis of x at an angle whose tangent is 0.5014. [In the approximate case the ellipses are
72 + 22 = const., and the major axis is inclined to the axis of x at
an angle tan1$.]
5. The question may be solved in general terms by putting YON = 0, XOM = 0, and replacing the probable errors 1.22 and 1.50 by a and b respectively; then the ellipses (2) are,
. . (6)
equation (3) becomes
,aye + (x  ,y 0)2 = C. titan 
7 () 
a2 + tan Bx=?rbtan 8 = 0 2 . 
(8) 
tan tan 0 X = = 62' + a2 tan28 

1 1 tan20 

_ e2 a2 + b2 . . 
(9) 
tan 0_e2 _ 
(10) 
. . tan 6 b2 
If c be the probable. error of the projection of p's whole motion on the plane of xz, then
0 = a2 tan2 0 + b2,
which is independent of the distance of p's line of motion from the
axis of x.
Hence also
tan 0 = a2
tan 0 b2 .
. (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