MICROMETER. 



209 



Double which they subtend when the semilensea are at f, the 

 ^Mi- other extremity of the scale, we have an instrument 

 will measure, with the utmost accuracy, all in- 

 termediate angles. 



In constructing this micrometer for astronomical 

 purposes, the semilenses may be made to move only 

 along a portion of the axis Of, particularly if the in- 

 strument is intended to measure the diameters of the 

 sun and moon, or any series of angles within given 

 limits. By increasing the focal length of the semi- 

 lenses, or by diminishing the distance between their 

 centres, the angles may be made to vary with any de- 

 gree of slowness, and of course each unit of the scale 

 will correspond to a very small portion of the whole 

 angle. The accuracy and magnitude of the scale, in. 

 deed, may be increased without limit ; but it is com- 

 pletely unnecessary to carry this any farther than till 

 the error of the scale is less than the probable error of 

 observation. 



Let us now examine the theory of this micrometer, 

 and endeavour to ascertain the nature of the scale for 



uring the variations of the angle. For this pur- 



CCCLXXVI. poae, let LL, Fig. 1, be the object-glass, which form* 

 Fig. 1. an inverted image, m m, of the object M N, and let the 

 semilenses AB, having their centres at an invariable 

 distance, be interposed between the object-glas* and 

 its principal focus, in such a manner, that their cen- 

 tres are equidistant from the axis Of. Now, it is ob- 

 vious, that the size of the image M M is proportional to 

 the sice of the object MN ; and, a* the angle subtend- 

 ed by MN" depends upon its ize, the magnitude of 

 the image m m may, in the case of small angles, be as. 

 sumed as a measure of the angle subtended by M N'. 

 A* the ray* which proceed from the point M, are all 

 converged to M by meant of the lens LL alone, the 

 ray 6 A, which passe* through the centre of the semi- 

 lens A, must of course have the direction b m ; and, 

 as it suffers no refraction in pawing through the cen- 

 tre of A, it will proceed in the same direction 6 A m, 

 after emerging from the ennilen*, end will era** the 

 axis at F. For the same reaiBOS, the ray c B, pro- 

 ceeding from N, and peering through the centre of B, 

 will cross the axis at 1 , as it advance* to n. If the 



of F from A and B happens to be equal to 

 the focal length of the lenses A and B, when combin- 

 ed with LL, <litinct image* of M and N will be form- 

 ed at F, and they will appear to touch one another ; 

 and the line in n Iwing the sue of the image that would 

 have been formed by the lens LL alone, will be a 

 measure of the angle subtended by the points M.'N. 

 If the point F, where the lines A m, U n cross the 

 axis, should not happen to coincide with the focus of 

 the lenses A, B, when combined with LL, then let 

 this focus be at F', nearer A and B than F. Draw 

 the lines AF'm, BF'n, then it is obvious, that if the 

 angle subtended by M N were enlarged, so as to be 

 represented by ' wf, instead of n m, or so that the lens 

 LL alone would form an image of it equal to ' M', the 

 point of interaction F would coincide with the focus 

 F ; so that, in every position of the lenses A, B, with 

 rwpect to LL, the points M, N may always be made 

 to subtend such an angle, that when they are placed 

 before the telescope, the points F, F' will coincide, 

 aad consequently the images of the points M, N will 

 be distinctly formed at F', and will be in contact 

 Whenever this h opens, the space *wt will be a mea- 

 sure of the angle tfa* subtended by UN. Hence H 

 follows, that whatever be the position of the semi- 

 lenses A, B, on the axis Of, the rari b A, c B, which 

 TOL. xnr. nar L 



pen through the centres of the semilenses, will cross Double 

 the axis at some point F, corresponding with the focus Image Mi- 

 of rays diverging from M, N, and will mark out the c "~ 

 size of the image n' m', and consequently the relative 

 magnitude of the angle subtended by the two points 

 M, N. 



From the equality of the vertical angles A F' B, 

 n' F' m', and the parallelism of the lines AB, ir* wf, we 

 shall have n! m' : AB=/F' : GF' ;[and calling /F'=6, 



F 6 

 and considering that GF'=p 7, F being the focal 



F6 



length of the semilenses, we have n' at' : AB=6: _. , 



x 1 -4-f> 



and consequently n' "' = AB + ^? Now, call- 

 ing AB=2, F=10, and 6=1, 2, 3, successively, we 

 shall obtain 



. ,*.. + !-. 



' m' = 8 



=2.* 



=2.6 



from which it appears, that when b is in arithmetical 

 progression, the angle n' m' varies at the same rate, 

 and consequently the scale which measures the varia- 

 tions of the angle, subtended by the centres of the two 

 images, is a scale of equal parts. 



This instrument undergoes a very singular change, 

 when constructed as in Fig. 8, so that the semilenses Fig. i 

 are outermost and immoveable, while another lens, 

 LL, is made to move along the axil (.< /. In this case, 

 a double image U formed as before, but the angle sub. 

 tended by the centres of the images never suffers any 

 change during the motion of the lens LL along the 

 axis of the telescope. If the two images are in con. 

 tact when the lens LL is close to the semilenses, tliey 

 will continue in contact in every other position of LL; 

 but the magnitude of the images is constantly in creas- 

 ing; during the motion of LL towarili /i the principal 

 focus of the semilenses. The reason of this remark- 

 able property will be understood from Fig. S, where 

 M, N, are two objects placed at such an angle, that 

 the ray* passing through the centre* A, B, of the semi- 

 lenses, cross the axis at F, the focus of the combined 



for rays divergent from M and N. In this i 

 distinct images of M and N will be formed at F, and 

 will consequently be in contact. If the lens LL is re. 

 moved to the position L' L', the rays M IN, N n, which 

 are incident upon it at the points m and M, having the 

 same degree efconvergency as before, will be refract- 

 ed to F', the focus of the combined lenses for rays di- 

 verging from MN. Two distinct images of the ob- 

 ject will therefore be formed at ', and these images 

 will still be in contact In like manner, it may be 

 shewn, that whatever be the position of the lens LL 

 between G and /", the rays MJ\ N/J will cross the axis 

 at a point coincident with the focus of the combined 

 lenses, and will there form two images always in con. 

 tact Hence it follows, that though the magnifying 

 power of the instrument is constantly changing with 

 the position of the lens LL, yet the angle subtended 

 by the centres of the two images never suffer* the 

 least variation. 



The application of the divided object-glass micro, 

 meter to a telescope for measuring distances, and to a 

 coming- up glass for ascertaining whether a chip is ap* 

 2 D 



