40 6 Mr. atwood’s Theory for ike Menfuration 
is proportional to the fine of FI and the cofine of DL jointly ; 
confequentiy the fine of FI being the fame, fin. { ED is propor- 
tional to the cofine of DL ; this will lead to the reafon why in 
enumerating (art. 4.) the * conditions which limit the magnitude 
of the obferved arcED,the pofitionof the fecondaryKP, in refpe£t 
of the point of interfeftion Q^jmd of the fixed fecondary KO, 
was annexed ; for it will appear, that every thing elfe being th« 
fame* the magnitude of the arc ED will depend on the pofition 
qf the fecondary KP, whether it be 011 one fide of the fixed fe- 
condary KO, or on the other, the angles PKO,^KO, being equal. 
Having fet off 0 /= 0 P draw the fecondary K p interfering 
the parallel FIMU in the point U ; and through B and U draw 
the arc of a great circle BUW ; take UW = BU ; and through 
D and W draw the arc of a great circle DW : then by the 
conftrurion and demonftration in art. 6. the angle fubtended 
by the obferved objects will be meafured by the arc DW* and it 
will be eafy to fhew, that DW is not equal to DE, except in 
two extreme cafes ; that is, when the fixed plane of reflexion 
DFB is either coincident with the primitive fecondary KO or 
perpendicular to it. Through the points F and U draw the 
arc of a great circle VFU* and from D draw the arc DV. 
perpendicular to VFU : fince f the fines of half the arcs 
DE, f>W, are in a proportion compounded of the proportions 
of the fine, of IF to the fine of FU, and of the cofine of DL 
to the cofine of DV, the fines of IF, FU, being equal by the, 
conflruclion, the fines of half the arcs ED, DW, will be in fhe 
fame proportion with the cofines of DL and DV, which are 
evidently unequal ; confequentiy, the fines of half the arcs 
DE, DW, and therefore the arcs themfelves*. muft be unequal. 
i i. The angles PKO, OK^, remaining equal, when the 
fixed plane of refleftion BFD (fig. 4.) is coincident with the 
* Compare fig, 2. f Supra, 
fecondary 
