KiRSTiNE Smith 
85 
By eliminating the r's from the second of the normal equations we get an equation 
identical with (120), which shows that we have one more element than we can 
determine. 
From (120) and (121) we are however able to find 
(sin?«Aa — cos?iA6) and Am; we get 
sin MAa — cos « A6 = — 7,\ S {(mn — nu r .) (sin mAcc,- — cos mA?/,)} 
and Am = — ^ S {{m^ — r,) (sin vAxf — cos u/S.y,)}, 
where = 2 {r,} and = ^ ^ 
For a point of the adjusted line corresponding to we find, according to (119), 
Pg = sin vAxg — cos mA?/, = sin wArf — cos uAb — Tj,Au. 
The standard deviation of is seen to be the standard deviation of the position 
of the adjusted point (x^, ^/j,) in the direction at right angles to the line. 
We find 
T,! = — 2\ S {['"^ ~ '"i*"' ~ ('"'1 ~ ^i)] (^^^ uAx.; — cos nAy,)} 
iV (?^2 — 'yy^i) 
and o-^, = T7 (a sin^ u + y cos^ m) ] 1 ' ^ 
iV ( — m 
This standard deviation is quite analogous to that obtained for an adjusted 
ordinate when the abscissa is errorless and gives the same indications for the dis- 
tribution of the observations. 
For cr„ we find 
CT^ (a sin^ u + y cos^ u) 
again emphasising that the standard deviation of the rs ought to be a maximum 
to give the best determination of the line. 
In conclusion I should like to express my thanks to Miss H. Gertrude Jones 
for the care she has devoted to the preparation of the diagrams in this paper. 
