276
OBJECTIVES, THE TESTING OF COMPOUND
17, 17' and /. Thus tho focal length, and tho positions of tho foci, may bo found by measuring tho distances £j, £/ ; £2, £»' 5 £s> £s' of throo pairs of conjugate points from a iixod point.
(b) If N, N' aro tho nodal points of tho objootivo,
ij'-ij=2/+rf,
whoro (Z~NN'. Thus in cases where d is known or may bo neglected it is suiliciout to dotonnino tho distances £lf £/ ; £a, £/ of two pairs of conjugate points. This method is most easily carried out by keeping tho positions of tho object and imago pianos fixed and moving 0 along tho axis into tho two positions for which tho image falls in tho latter piano. Tho distances of object and imago from Q aro then measured for tho two positions, Q boing kept fixod relatively to 0.
(o) Tho distance 1) botwoon conjugate points is given by
I) •-= -x + x' l-2/f (/.
Now it follows from tho liquation ;)•/'— -•/* that -*•(•*' in stationary when -.c-.r'-- I/. Tho negative sign gives tho two nodal points, whilo tho positivo sign gives two [xiintH on tho opposite sides of tho foci from tho nodal points and at distances from the foci equal to /. in this oaso
D ~4/+ d.
Thus if d is known or may bo neglected, tho fooal length may bo found by moving tho objootivo until tho distance botwoon object and imago ia a minimum.
It may bo noted that in method (ii.j («) Ihe focal length may bo found by measuring £, £' for ono pair of conjugate* points if tho positions of tho foci F, .F' aro known, that is, if 17, 17' aro known. A knowledge* of / then enables ono to determine tho distance d between tho nodal points by means of methods (ii.) (/>) or (r).
The methods in group (ii.) can be carried out most conveniently on an optical bench. They aro specially applicable to objectives of medium fooal length. Tho chief objection fco employing them in. the ease of long focus objectives is tho great length of wpaeo required, tho minimum distance between conjugate points boing, as we have soon, approximately four times the focal length.
(iii.) Magnification Methods.—There aro a number of methods of determining focal length which depend on tho measurement of tho lateral magnification. Tho most important of fchoso aro as follows.
(«) The magnification is measured for two positions of tho object, the system under test being kept fixed. Then, if xlt xz arc the distances of the object from the first principal fooal plane and ma, m3 aro tho magnifications
for the two positions, it follows from equation '•2) above that
3V:         /• 2     W, ^
Thus               /- i ^
It is only necessary, therefore, to measure the distance a^-tfj through whicli the object has boon moved, in addition to tho magnifications. Tho focal length may also be found by kctyping tho system fixed, moving the, image piano a distance x^ - £a', and measuring tho corresponding magnifications. In this case
•mj,   a;a' -• - mj.
Thus
A Himplo diri'ot/ rending method bnflcd on this relation can lit* employed in (hiding llio fooal k-iigth of a (lamcra ICIIH. Tlio imago nf a vt-ry tliHltmt olijcnt !H fii»ufuu>(l on tho grontul-gloHH mu-con »f tlit* canu-ra and llio Horooii !H thon racked lia<^k (hrougli a diHlaucto d until thi" imago <if a scialc, nuumU'cl near (lie lens in ii piano at right nnglcn to tho option! axiH, ia focuaat'd nn it. The length of the portion of tho Htialo, whotm imago HOH lielwoen two rnarkn separatod by a distance d on tho screen, IH tlien tho focal length of tho lonH.
(b) Tho distance fj tif the object from a fixed point Q (Fig, 2) on the objective is measured and the objective is then reversed and moved along the optical axis until the same magnification is obtained, the object and image planes being thus interchanged. Tho diHtmico of the original object from Q is now measured. Since tho positions of V and F' aro interchanged owing to the reversal of tho systom, it follows that
Tlioroforo
*, * - D + - 4 *'
mf
If, in the second ^losition of tho fiystem, tho distance £/ of the original imago plane from Q is measured, then
.-. J       m- 1/m'
(r.) The magnifications ml and w8 aro measured for two values l)t and I)a of tho distance between the object and imago planes. [f, for example, the distance is increased by an amount dt wo have
D3
•= d ~~ ( - xt 4- xa") • - ( - a?t f