408 Mr . atwood’s "theory for the Merifuration 
muft bs lefs than 90°. If BM be lefs than a quadrant, an 
obfer vation by two reflections may be taken when the radius 
CP is directed to any point in the circumference of the plane of 
motion. When the arc BM is greater than a quadrant, two 
other limits will be produced in the circumference of the plane 
OCP ; while the radius CP is between thefe limits, no obfer- 
vation by tw~o reflections can be taken : thefe limits are con- 
ftruCted thus (fig, 6.). BM being greater than a quadrant, with 
the pole B and diftance BI equal to a quadrant, deferibe the 
arc of a great circle I / interfering the parallel FIM in the 
points I and /: through I and / draw the fecondaries KY, KZ: 
while the radius CP is between Z, and Y no obfervation can be 
taken by two reflections. If BIE, B/E, be drawn equal to a 
femi-circle, and DE joined, then DE will be the meafure of 
the limiting angle which can be obferved by this con ft ruction, 
either on one fide of KO or on the other ; and becaufe, by the 
principles of trigonometry, the arcsBD andDE are in the fame 
great circle, BDE being a femi-circle, we fhall derive from the 
conftruCtion this conclufion : the difference between i8o°and 
double the angle of incidence on the fixed fpeculum, will be a 
limit which terminates the angle obferved by two reflections in 
every cafe, when the arc BM is greater than a quadrant 
14. In any given example formed on the principles which 
have been demonftrated (fig. 2.) for the eftimation of angles 
by two reflections, three of the four quantities neceffary to 
determine the refult muft conftantly be the fame, while the 
fourth, that is, the arc OP, varies with the magnitude of that 
angle fbfe tended by the objeCls obferved : the different magni- 
tudes of thefe three given quantities will caufe a great variety 
of properties in conftruCtions which depend on the general 
Thu's termination of the angle which can be obferved by two reflexions may 
happen while the obferved angle is increafing or decreafing during the revolution 
of the index in the plane of motioji, 
