OPTICS. 



OPTICS. 



eonndention are tboM which are directed nearly along the axis, 

 and whkh therefore fell exceedingly nearly perpendicularly uii th- 

 reflector. 



Let D B B represent the section of a spherical reflector made by a plane 

 paving through Hi axis, c its centre, A the focus of incident ray, A D an 

 incident ray, A D c is the angle of incidence. Make the angle a D c = 

 A n r, then a D c ii the angle of reflection, and if the point of incidence 

 D were infinitely near to the point B in the axis, then all the reflected 

 raya of which the incidence wai nearly perpendicular would converge 

 lie latter would then be the focui conjugate to A, for if rays 

 diverged from a they would after reflection evidently converge to A. 



Now, if a straight line ai c D bisect an angle of a triangle, aa the 

 angle A D a, it will divide the baw into segment* A c, c a proportional to 

 the adjacent sides A D, D o (' Euc.,' book \i.), that is, A c : oo : : AD: Da; 

 but when D is infinitely near to B, we may write ABandsa instead of AD 

 and DO, in which case we should haveAO:ca: : AB :Ba. Let AB-- A, 

 aa--&', and the raditu CB = r; then AC=A r; cu=r A'; whence 

 A _ r : r - A': : A : A', or A' (A-r) = A (r-A')i therefore 2 AA' = r 



112 



(A + A 7 ), which may be also written in the form _ + - = -. Wi- 



A A r 



should have precisely the game investigation if we had supj>oaed rays 

 as B D to fall on the convex side converging to a focus A ; but being 

 reflected in the direction DC, they would appear to diverge from the 

 conjugate focus a : hence the above formula applies to two eases, 

 namely, when diverging rays fall on the concave surface, or converging 

 rays on the convex surface, of a spherical reflector. 



Example 1. A candle is placed before a concave speculum at a 

 distance of 3 feet from it : what will be the distance of its image from 

 the same, the radius of the speculum being 2 feet ? 



Here we have given A = 8 feet, r = 2 feet, and to find A' we substitute 



112 



these numbers in the general formula _ + _ = -, which thus becomes 



A A r 



!+.!_ = ?= 1 ; whence --, and therefore A'= - = H foot; the image 



3 A' 2 A' 3 



will consequently be 1 foot 6 inches in front of the speculum. 



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 

 fiifil a proper formula from the proportion we have established 

 namely, AC:ca::AB:sa. Let A c = p, c a = p' ; then A B = r + p, 

 B=r-p',whence/> :'/>' : : r+p : rp', l or p (r-p') =f>'( r + p), therefore 



r(ff') = Zff', consequently i 1=?; thus, in the example given 

 p p r 



above, we find (since p=l and r=2) ( 1 = 1, or - = 2; therefore p' = 



-, 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 



A 



small and may be rejected, in which case we have = - or A' = -, that 



A r 2 



is, parallel incident rays are made after reflection to converge to F, the 

 middle point of the radius c n. Hence the focal length of a spherical 

 speculum is one-half that of its radius. 



In examining the formula for the positions of the conjugate foci, 



112 

 namely, - + = -, we find that when A = r we also must have A' 



A A' F 



= 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 Jig, 1), A 



being then greater than r, is less than -, and therefore must be 

 A r A 



greater than - or A' is less than r, and as A increases to greater magni- 

 tude*, A' accordingly diminishes, until A becomes infinite, when A', on 

 we hare seen, becomes - : hence, whilst A moves on the left indefi- 

 nitely from c, the other focus moves on the right from o as far as the 

 principal focus r. 



With respect to the images formed by concave specula, let AO 

 represent a small object at A, the line A o being perpendicular to A c, 

 join o c, then o c I will be the axis of the speculum when o is con- 

 sidered the focus of incident rays, and its conjugate focus y can be 

 found by the preceding formula : hence a g will be the image of A a, 

 it* position is evidently inverted, and it is easy to see that the line ay 

 is very nearly perpendicular to c B ; and by similar triangles, the linear 

 dimensions of the image ay are to those of the object A o as c y : c a, 



or as p' : p. Now the formula 1 I = - gives f - = -1 ; hence 



?' P r f 2p + r 



the 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 : 2A r; on the contrary, 

 if the object be placed between the centre and principal focus, as 

 at ag, then AO would become the image; for AO : ay ..pip'; but 



f - = 5-,= 5 r ! therefore A o : a,'g : : r : 2 A' r ; which shows 



that the image is then greater than the object, or magnified. From 

 Hi.- 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 how much it will 

 appear diminished in its image, with respect to iU linear, superficial, 

 and solid dimensions. 

 Here r=2, A=12, 2A-r>=22; therefore 



for linear dimensions Image : Object : : 2 : 22, 



that is : : 1 : 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 collectedby a concave speculum in its principal focus (or burning- 

 point) P. 



Example 3. To find how much an object will be magnified by the 

 same speculum, when placed 1 foot 6 inches in front of it. 



Here A' = 14, r=2, 2 A' r=l; therefore in linear dimensions the 

 ratio is as 2 : 1 ; 



in superficial as 4 : 1 ; 

 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, &c. 



rig. 2. 



Let A D be an incident ray near the axis A c, join c D and produce 

 to c ; make the angle of reflection CDC equal to the angle of incidence 

 A D c, and produce the reflected ray T>e to meet the axis in a ; th 

 when D is infinitely near B, a is the focus conjugate to A. The same 

 figure would equally apply if we had supposed rays B D converging to A 

 to fall on the concave surface, for since the angles ADC, c D e, a DC, 

 c D E 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 D a is bisected by the straight 

 lino D c, it follows (Simson's ' Euc.,' book 6) that Ac:ca::AD:oa 

 (and D being supposed infinitely near to B in order that the rays may 

 be incident nearly perpendicularly) : : A B : B o. Let A B= A, a B = a!, 

 cu=r, CA=p, ca=p', then we have 



p : p' : : A : A', or r + A : r A : : A : A', hence 

 A'(r + A) = A(r A'), 



112 



therefore 2AA'=r (A A') whence-, jj = - . Again the same pro- 

 portion p : p' :: A : A' may be written p : p' : : p r : rp' ; hence 



112 

 P ( r ~P r )~P 1 (f~ r ) therefore r (p + p') = 2pp' whence - + = - 



If we suppose p=r, we findp'=r, which shows that the foci are 

 together at B, and as p increases, p' diminishes, until p becomes infinite, 



when p' = ^ ' showing that a will then reach the principal focus r. 



Hence, in general, the principal foci move in contrary directions, and 

 meet both at the centre and circumference. In the formula just given 

 one of the conjugate foci lies between the principal focus and the 

 surface of the speculum ; while in the first set, one lay between that 

 point (F) and the centre. A single formula, however, as the reader 

 may verify, will include all cases, whether of concave or convex specula, 

 or divergent or convergent pencils, provided we admit the use of 

 negative quantities, and interpret the negative sign to mean that the 

 distance affected by it must be measured in a direction contrary to 

 that in the figure on which the formula was framed. 



With respect to images, if A o be the object and g the focus con- 

 jugate to o, then ay will be the image of A o ; and conversely, Hag 



