4i6 
. Di 
fin. DL 
7 . 2 
% m — *7i n 
Mr. atwood’s Theo'ty for the Menfuraiton 
7 
v' I — />-• X ^ »4 — »r-n z v' 
k 2 . 2 2 .2 , 2 ^ 2 2 
^i-,y 
p m n — p n m s ^ 
* •■ 2 a — p* fr? 2p % rn l tr — m 1 r?p z fz+zijtf'pn X E l — p z x v' 1 — 4 x 4 i — r? 
i-4/4 
and the ftp are of the cofine of DL 
I — ep z — n z + m Z K 2 -f-p i m z — lp z rr?n z + w 2 n z p~ 4 ± inCpn X 4 I — p z X ^ l-i 2 X^l — 4 
A _ ~ TT2 ’ ' r~ 
I -Sp 
The fine of IF was fhewn to be 2spx\fi—s z f, and its 
fqnare = 4^* x 1 — s'p 1 ': moreover, it was demonftrated in 
art. 9. that as rad. 1 : cof. DL 2 :: fin. Ib 2, : im. f ED 2 which 
gives, by fubftituting the values of col. DL 2 and lin. IF 2 , and 
multiplying the cof. DL 2 into lin. IF 2 , lin. § ED 2 = 
p“ X I — 4/4 — ri z p 2 t?r - 1 p~ m z r z + mVrf±2m 2 /)»X f l-j z x ^ i —p 1 x L i — 4 
and the cofine of § ED 2 = 
4 .y x l-rp -m z + fnrr + p*m 2 -lp z m~u+m z n z s l p z ±:iiri pn X V i-4x Vl-p z X 41-/4 
finally the cofine of ED is therefore = 
84/ 
X l—i^p^—m 2 4* «4« 2 -{-p z n z —ip z m z n z -\-m z n zz p 2 ±^iTrfpn x |/ i-4 x / i— p z x */ i—tf. 
22. The particular cafes inferred from the geometrical con- 
flruCtion may be compared with this analytical value of the cofine 
of ED, or of the angle fubtended by the obferved objects. If 
j= 1 and « = 1, by fubftituting 1 for s and «in the expreffion 
juft found, we {hall have the cofine of ED = 1 — 8y> 2 4- 8y> 4 , which 
is the cofine of an arc four times greater than that of which the * 
fine ~p. This anfwers to the properties of hadley’s inftrument, 
in whichKFor the inclination of the reflecting planes totheplane 
of motion is 90°, and its fine = 1 ~s: moreover, in hadley’s 
inftrument, the fixed plane of reflection at the unmoved fpecu- 
lum is parallel to the plane of motion, and therefore perpendi- 
cular to any fecondary of that plane ; its inclination to any fe- 
condary 
