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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,

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 tan-1\$.]

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