4i 6 Mr. atwood’s Theory for the Menfuration 
fin. DL = and 
¥ 
im. DL ! 
a 7ft 
:* — n ?n z - p 2 m 2 ^2p % m z h z -r> i 2 n z p z s' + 2 ?ri t pn'* ^ \ -p z x a/ i - s z X *S I -n 
l 
and the fquare of the cofine of DL 
1 - fp*— n z + /ft V -f p 2 n? - 2 p z m z r z + m z *p zz ±2m z pn X ^ I - p z X ^ l -j 2 X ^ i - n 
l ~*F 
The fine of IF was fhewn to be 2 spy v/ 1 -s 7 p\ and its 
fquare ^ 4 s'p z x 1 — spf : moreover, it was demonftrated in 
art. 9. that as rad. 2 : coi.DL 2 :: Fin. IK : im. I ED 2 which 
gives, by fubftituting the values of col. DL 2 and tin. IF 2 , and 
multiplying the cof. DL 2 into tin. IF 2 , tin. \ ED 2 = 
4 2 p] x I - i z p z - /ft 2 4 " ” 2 p z n z i z + n z ) zz p z ±: 2 ?n z p 1 ;.X ^ I - j 2 X ^ i —p“ X ^ i -ft 
and the cofine of 4 ED 2 = 
X-\ z p 2 x l-s z p z — nr nr n z p z m z — 2 p z .-/ z / z -\-n z n z s'p z ±: 2m pn X V l-j 2 x V i-p~ X Vi-ft 
finally the cofine of ED is therefore =r 
%-% z p z X i—i 2 p z -m 2 -f n z n z J rp z r; z - 2 p z n z r z A^tnr 2 s z p z =£.2n z pn x i-J 2 X y'" i-/>* x V 1-ft 2 
22. The particular cafes inferred from the geometrical con- 
frruftion may be compared with this analytical value of the cofine 
of ED, or of the angle fubtended by the obferved objects. If 
s—i and n~- 1, by fubftituting i for s and n in the expreffion 
juft found, we fhall have the cofine of ED =■ i — 8 p z -f 8 p\ which 
is the cofine of an arc four times greater than that of which the 
fine — po . This anfwers to the properties of hadley’s inftrumenr, 
in whichKFor the inclination of the reflecting planes to the plane 
of motion is 90 °, audits 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 
