VOL. XXXVII.] I'HILOSOPHICAL TRANSACTIONS. 487 



MFA double the angle hfa or mfh; consequently rm is equal to ra, and the 

 angle mra double the angle hra or mrh. In like manner, the point n is also 

 in the plane of the scheme, the line rn equal to rm, and the angle mrn double 

 the angle MRi oriRN; subtract the angle mra from the angle mrn, and the 

 angle arn remains equal to double the difference of the angles mri and mrh, 

 or double the angle hri, by which the surface of the speculum de is reclined 

 from that of bc, and the lines ra, rm, and rn are equal. 



Corol. 1. — The image n will continue in the same point, though the two 

 specula be turned together circularly on the axis r; so long as the point a 

 remains elevated on the surface of bc, provided they retain the same incli- 

 nation. 



Corol. 1. — If the eye be placed at l, the point where the line ap continued 

 cuts the line gk ; then the points a and n will appear to it at the angular dis- 

 tance ALN, which will be equal to arn; for the angle aln is the difference of 

 the angles fgn and gfl; and fgn is double fgi; and gfl double gfr; and 

 consequently their difference double frg or hri; therefore l is in the circum- 

 ference of a circle passing through a, n, and r. 



Corol. 3. — If the distance ar be infinite, those points a and n will appear at 

 the same angular distance, in whatever points of the scheme the eye and specula 

 are placed: provided the inclination of their surfaces remain unaltered, and 

 their common section parallel to itself. 



Corol. 4. — All the parts of any objects will appear to an eye viewing them by 

 the two successive reflections, as before described, in the same situation as if 

 they had been turned together circularly round the axis r, keeping their res- 

 pective distances from one another, and the axis, with the direction hi, i.e. 

 the same way the second speculum de reclines from the first bc. 



Corol. 5. — If the specula be supposed to be at the centre of an infinite sphere: 

 objects in the circumference of a great circle, to which their common section 

 is perpendicular, will appear removed by the two reflections, through an arch 

 of that circle, equal to twice the inclination of the specula, as is before said. 

 But objects at a distance from that circle will appear removed through the similar 

 arch of a parallel; therefore the change of their apparent place will be measured 

 by an arch of a great circle, whose chord is to the chord of the arch equal to 

 double the inclination of the specula, as the sines complements of their res- 

 pective distances from that circle, are to the radius; and if those distances are 

 very small, the difference between the apparent translation of any one of these 

 objects, and the translation of those which are in the circumference of the 

 great circle aforesaid, will be to an arch equal to the versed sine of the distance 



