Or let the prisms PR, RS, be placed with their re fracting angles P, S, turned from one another as in Plate CCCLXXV. Fig. 21, the refraction of the prism PR Mill transfer the image of the sun from ON to w e, and the refraction of the prism RS will transfer the image ON to 2w2e, the two images 2w2e w e, touching one another externally at the point 2e w. Let ECN, \VCO, be the axes of the p ncils of rays proceeding from the two extreme limbs of the sun, and N, 0, the points where the images of the tubtern and western limbs would be formed by the object-glass. .‘ere it not for the retraction of the prisms ; the ray EFR, which belongs to the axis ECN, and is refracted by the prism RS to 2e, undergoes the refraction NH 2e ; and the ray WGR, which belongs to the axis WCO, and is refracted by the prism PR to SU uudergoes the refraction OR w. Now NC 2e, part of the angle measured, is to NH 2e, the refraction of the prism RS, as R w to C 70; and OC w, the other part of the angle measured, is to OR w, the refraction of the prism PR, in the same ratio ot R w to C w : therefore OCN, the whole angle measured, is to ORN, the sum of the refractions of the two prisms, as R w to C w; that is, as the distance of the prisms from the focus of the object-glass to the focal distance of thc.object-glass.
When the prisms are placed in the manner repre sented in Fig 20, the point e of the image w e is illu minated only by the rays which fall on the object glass between A and F. and the point 2w only by the rays which fall on the object-glass between B and G. Now the angles CRF, CRG, equal to the refractions of the prisms, being constant, the spaces FC, CG, will increase in proportion as the distances RF, RG. in crease, and the spaces AF, G B, diminish as much ; and therefore the images at the point of mutual con tact, e 2 w, will be each illuminated by half the rays which fall on the object-glass when the prisms are placed close to the object-glass ; but will be enlightened less and less, the nearer the prisms are brought to the focus of the object-glass.
But when the prisms arc placed in the manner shewn in Fig. 21. the images at the point of contact, as the prisms are removed from the object-glass towards the eye-glass, will he enlightened with more than half the rays that fall on the object-glass, and will be most en lightened when the prisms are brought to the focus itself ; for the point 2e of the image 2w2e will be en lightened by all the rays EE that fall on the object glass between B and I', and the point w of the image w e will be enlightened by all the rays WW which fall on the object-glass between A and G. But the differ ence of the illuminations is not very considerable in achromatic telescopes, on account of the great aperture of the object-glass ; as the greatest space FG is to the focal distance of the object-glass, as the sum of the sines of the refractions of the prisms is to the radius.
There is a third way, and perhaps the best, of plac ing the prisms, so as to touch one another along their sides which are at right angles to the common sections of their refracting planes. In this disposition of the prisms, the images will be equally enlightened, namely, each with half the rays which lall on the object-glass, wherever the prisms be placed between the object-glass and eye-glass.
From what has been shewn, it appears that this in strument, which may be properly called the prismatic micrometer, will measure any angle that does not ex ceed the sum of the refractions of the prisms, except ing only very small angles, which cannot be taken with it, on account of the vanishing of the pencils of rays at the juncture of the two prisms near the focus of the object-glass ; that it will afford a xery large scale, namely, the whole focal length of the object-glass for the greatest angle measured by it ; and that it will ne ver be out of adjustment ; as the point of the scale where the measurement begins (or the point of 0) an swers to the focus of the object-glass, which is a fixed point for celestial ohjects, and a point very easily found for terrestrial objects. All that will be necessary to be done, in order to find the value of the scale of this mi crometer, will he to measure accurately the distance of the prisms from the focus, when the instrument is set to measure the apparent diameter of any object sub tending a known angle at the centre of the object glass, which may be easily found by experiment, as by measuring a base, and the diameter of the object ob served placed at the end of it, in the manner practised with other micrometers : for the angle subtended this object ‘vill be to the angle suuended by a celes tial object, or very remote land object, when the dis tance of the prisms from the principal focus is the same as it was found from the actual locus in the terrestrial experiment, as the principal focal distance of the object glass is to the actual focal distance in the said experi ment.
It will, I apprehend, be the best way in practice, in stead of one prism to use two prisms, refracting con trary ways, and so divide the refraction between them (as represented in Fig. 20. and 21.) Achromatic prisms, each composed of two prisms of flim and crown glass, placed with their refracting angles contrary ways, will undoubtedly be necessary for measuring angles with great precision by this instrument.