84 Choice in the Distribution of Observations 
It rests to compare tan u and — , we find 
tan = - — /xoa - /u-ao 
r20 ^ril I ^20 
^'-02 ^'-20 
^■20. 
2 4„2^ I 
H r (/^02^^20-^n)t • 
.(119), 
The factor in curled brackets is hence positive and we have tan w > or < — - 
according as ' ix^^ > or < 0, 
we have thus proved that 
^ - tantt- ^ . 
t^to Mil 
(4) In order to find the standard deviations of the constants of the line we 
shall express the observations, the standard deviations of which are V aa and 
V ya, by a parameter /• to get an equation for each observation. 
Suppose Xi = a + Ti cos u, 
111 = & + r.i sin u, 
and suppose we have a good approximation for a, b, u, r^, r2 from which is 
calculated x and // corresponding to the observations. The diflerences between 
observed and calculated x and y can then be expressed by 
Ax,- = A« — sin u . Am + cos u . Ar;[ 
A?/; = A6 + Ti cos M . Am + sin u . Ar^- ) 
and we can carry out an adjustment, A«, A6, Am, Ar^, Afg ••• A/\v being the 
elements. 
The noi'mal equations are : 
1 < . N ^ „ , , sinw . cosM , cosm . 
- i, ic.-l = - Aa + 0 . A6 - r,- Am + Ar, + . . . + Ar^, 
a a a a a 
1 , , o . V. , 1 cosM . sin?t Kinw , 
- 2 ?/, =0 . A« + - A& + i: r,} Am + Ari + ... + Ar^, 
7 7 7 7 7 
^, ( r sinw . cosM 11 
= -2|?-j} Aa+S{rj|-— ^ A& + :i-|/-[-| — - H -—J Am +ri - ^ j cosm sinitAri +... 
+ rv('- — ^ COSM sin mAj-v, 
' \7 a/ 
cos?t . sinM . 
__Axj + --A2/. 
COSM . sinM /I 1\ . . /pos^M sin'^wX , a a 
= Aa + A& + r, cos u sni mAm + 1 Ar, + . . . + 0 , Ar v, 
a 7 \7 ay \ a y J 
cos M , sin u . 
Aj:^- + Ayx 
a 7 
COSM . sinM /I 1\ - . „ . /cos^M sin-M\ 
= Aa H A6 + r V cos m sm mAm + 0 . Ar, + . . . + 1 Ar^ 
a 7 \7 «/ \ a 7 / 
Eliminating r^, ... r,v from the first and the third of these equations by means 
of the last N equations, we obtain 
S {sin uAxf — cos wA?/,} = N sin uAa - N cos uAh — S {r J Am (120), 
and 
[sinwAa;,: — cosuAy,]}= 'L{r.^ sin wAa — S{r;}costtA6 — S {r'f} Am...(121). 
