It being sometimes convenient to measure the distances of the foci from the centre instead of the surface of the speculum, it is easy to an a proper formula from the proportion we have established namely, A c ca :: A 13 : B a. Let A c = p, c a = p' ; then An = r + p, a n =r -p', whence p :'p' r+ p : r-p',or p (r - p') =p' (r + p), therefore 1 r (p-p') = 2pp', consequently 1 -- P -=-2 ; thus, in the example given r above, we find (since p = 1 and r=2) =1, or =2; therefore which is agreeable with the former result.
When the incident rays proceed from a point exceedingly distant (as the sun for instance), then A being very great will be exceedingly small and may bo rejected, in which case we have 1 rs or that r 2 is, parallel incident rays are made after reflection to converge to r, the middle point, of the radius c n. Hence the focal length of a spherical speculum is one-half that of its radius. ' lu examining the formula for the positions of the conjugate foci, 1 1 2 namely, - + - = we find that when A = r we also must have A' A A' r = r ; hence when the focus A is at c the centre, the conjugate focus a will be at the same point. If A move to the left of c (in fig. 1), A being then greater than r, 1 1 1 less than _, and therefore _ must be greater than ! or A' is less than r, and as A increases to greater magni tudes, A' accordingly diminishes, until A becomes infinite, when A', as we have seen, becomes : hence, whilst A moves on the left indefi nitely from c, the other focus moves on the right from a as far as the principal focus F.
With respect to the images formed by concave specula, let A a represent a small object at A, the line A a being perpendicular to a C, join a c, then acs will be tho axis of the speculum when a is con sidered the focus of incident rays, and its conjugate focus g can be found by the preceding formula : hence a g will be the Image of A a, its position is evidently inverted, and it is easy to sac that the line a y is very nearly perpendicular to c a ; and by similar triangles, the linear dimensions of the image ay are to those of the object • a as cg or as 2 : p. Now the formula - - - = gives e = r ; hence P' p r p 2p+rthe image (in respect to linear dimensions) is less than the object in the ratio of r : 2p+ r (or since p= A -r) as r : 2 A-r ; on the contrary, if the object be placed between the centre and principal focus, as at a g, then A G would become the image; for AO:ag::p: ; but r = = ; tnerewro A a'g : : r : 2 which shows r-2p' -r that the image is then greater than the object, or magnified. From the principles of geometry it follows that the surfaces of the image and object are as the squares of the linear dimensions, and the apparent volume, or bulk, as their cubes.
Example 2.-An object is placed at a distance of 12 feet in the axis of a concave speculum of two feet radius : to find bow much it will appear diminished in its image, with respect to its linear, superficial, and solid dimensions.
Here r=2, a=12, 2A -r=22 ; therefore for linear dimensions Image : Object : : 2 : 22, that is : : I : 11; for superficial do. the ratio is as 1 : 121, and for apparent bulk it is as 1 : 1331.
Heat being capable of reflection, like light, the rays of the sun may be collected by a concave speculum in its principal focus (or burning point) r.
Example 3.-To find how much an object will ho magnified by the same speculum, when placed 1 foot 6 inches in front of it.
Here ..a'=1,1, r=2, 2.1'-r=1; therefore in linear dimensions the ratio is as 2 : 1; in superficial as 4 : I ; and in cubical as 8 : 1.
Let us next consider the relation between the conjugate foci when diverging rays fall on a convex spherical speculum, which will also be the relation when converging rays fall on a concave speculum as will be evident by inspection of the figure (fig. 2). Employing the same letters with the diagram as before, c will be the centre, A the focus of incident rays, a of reflected rays, dc.
Let A n be an incident ray near the axis A C, join c n and produce to c; make the angle of reflection e n e equal to the angle of incidence A D c, and produce the reflected ray De to meet the axis in a ; then when n is infinitely near n, a is the focus conjugate to A. The same figure would equally apply if we had supposed rays ED converging to A to fall on the concave surface, for since the angles A n c, e n e, a n c, n a: are all equal, D a would then be the actual reflected ray and therefore a would be still the focus conjugate to A. Now since the external angle a D E of the triangle A n a is bisected by the straight line n c, it follows (Simeon's ` Euc.,' book 6) that Acioa::AD:Da (and n being supposed infinitely near to is in order that the rays may be incident nearly perpendicularly) ::Anin a. Let A n= A, a is =A', ou=r, c A=p, c a= p', then we have p: p' :: A : A', or r + A : r —A : : A : A', hence (r + A) = A (r 1 I 2 therefore 2a A'= r (A-A) whence z = . Again the same pro portion p : p' A : A' may be written p : : p- r : r-p' ; hence p (r-p')= (p-r) therefore r (p + p')= 2 pp' whence 1 1 = 7. • If wo suppose p=r, wo find p' =r which shows that the foci are together at n, end as p increases, p' diminishes, until p becomes infinite, when p' showing that a will then reach the principal focus r.