of the Angle fubt ended by Two Qb/effs, &c. 417 
condary therefore will be 90% and the fine of this inclination 
~ 1 — n by the problem. And fince p is the fign of half the 
inclination of the refledtors, the angle of which the cofine is 
1 — 8 p z 4- 8^ 4 will be twice the inclination of the refiedting planes, 
which is a property of hadley’s inftrument. In the analytical 
value of the cofine of ED, the laft term is affedted by two figns ; 
thefe depend on the pofition of the fecondary KP and the in- 
terfedtion Q in refpedt of the point O. If the fecondary KP 
or the index CP be on the fame fide of O with the interfedtion 
Q (fig. 2.), the fign of the laft term is negative: if CP and 
CMae on oppofite fides of O, the fign of the' laft term will 
be pofitive; and when DFK=r 0 or 180°, the whole term 
vanifhes, becaule in that cafe n = o. Alfo, if m~o 9 n=i 9 
s = 1 , or if p = 1 , the laft term vanifhes. When mzzs~ 5 , 
v 2 
KF = 45 0 : in this cafe, if n — 0 the conftrudtion will be that de- 
fcribed in art. 15. and the cofine of the obferved angle ED will 
equal 1 — 2 p 1 , the other terms vanifhing : andbecaufe 1 — 2 p z is 
the cofine of an arc double to that of which the fine ~ />, it 
follows, that the angle obferved will be equal to the arc 
defcribed by the index from 0, of which the fine of one' 
half is by the problem =: p. In every cafe, when n = o 9 
that is, when the fixed plane of reflection at the unmoved fpe-~ 
Culum coincides w T ith the primitive fecondary KO (fig- 2. and 
1 2.), the cofine of ED — . 1 — 8 j 2 /> 2 x 1 — s z ff — in 4 -p l nf. 
23. The fine of ED will be neceflary (art. 27.) to afcertain 
the variation of ED from the truth oecafioned by errors in 
the data ; to obtain fin. ED let 
l-fp^-rn 4 r}?n 4 ft rtf- %p l m'ri' 4 » 2 /2V/± irrpnp xVi-s l X’l/ i-p z X V 1 -n z 
zzd: then (art. 21.) from the value of col. ^ED we have 
fin. ED= ^sp x >/d x s/i - 4f/>V. When s is very finally 
and 
