of the Angle fubt ended by Two Objects, &c. 417 
condary therefore will be 90°, and the fine of this inclination 
~\~n by the problem. And fin Cep is the fign of half the 
inclination of the refleCtors, the angle of which the cofine is 
1 — 8 p 1 + 8/ will be twice the inclination of the reflecting planes, 
which is a property of hadley’s inftrument. In the analytical 
value of the cofine of ED, the laft term is affeCted by twofigns ; 
thefe depend on the pofition of the fecondary KP and the in- 
terfection Q in, refpeCt of the point O. If the fecondary KP 
or the index CP be on the fame fide of O with the interfe&ion 
Q (fig. 2.), the fign of the laft term is negative: if CP and 
Q> on oppofite fides of O, the fign of the laft term will 
he pofitive ; and when DFK = 0 or 180°, the whole term 
v^nifhes, becaufe in that cafe n~o. Alfo, if m~ o, n~i r 
s — 1 5 or if p ~ 1 j the laft term vanifhes. When ?n~s = — ~ 
^ 2 
KF = 45 0 : in this cafe, if n = 0 the conftruction will be that de- 
fcribed in art. 1 5. and the cofine of the obferved angle ED will 
equal 1 - 2p\ the other terms vanifhing : and becaufe 1 - 2p x is 
the cofine of an arc double to that of which the fine - p , 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^ 
that is, when the fixed plane of reflection at the unmoved fpe- 
culum coincides with the primitive fecondary KO (fig^ 2. and 
1 2.), the cofine of ED = 1 - 8 Ap z x 1 —sf- ~m j-pirf, 
23. The fine of ED will be neceffary (art. 27.) to afcertain 
the variation of ED from the truth occafioned by errors in 
the data ; to obtain fin. ED let 
I-sy-m * + ; 
"J: then (art. 21.) from the value of cofi |ED* we have 
fino ED ~ 4 sp- x \/ d x %/i — 4ypV. Wlien s is very final!. 
and 
