HARDWICKE'S SCIENCE-GOSSIP. 



267 



constructed for my own use. It is formed of two 

 pieces of wide J-inch board (as indicated by dotted 

 line across centre of figure) fastened together by cross 

 pieces screwed across the ends. At right-hand end 

 is a small platform for the microscope, raised \\ inch 

 (2 inches would be better) above the rest of the tray. 

 On it the instrument can be moved forward and 

 backward, but not sideways. The raised platform 

 allows the mirrors, when in their lowest positions, to 

 be set on a level with the lamp-flame, another im- 

 portant simplification of the problem we have in 

 hand. The tray can be very easily put together by a 

 little amateur carpentry. 



Having such a tray, we must next select the two 

 angles of slope for the tube of the microscope. In 



Fig. 167. 



my own case angles of 40 and 30° with the surface 

 of the table are found convenient. A thread with 

 two buttons attached, can be hung beneath the tube, 

 so that when either of the buttons just touches the 

 platform, the instrument will be sloped at one of the 

 two selected angles. 



The complication arising from the tilting of the 

 microscope will probably be the better understood, if, 

 in the first instance, we suppose the tube to be in the 

 horizontal position, and enquire how in that case we 

 could assign the right position to the lamp. Remem- 

 bering that any angle of incidence is equal to half the 

 angle formed by the ray with the axis of the micro- 

 scope (so long as the mirror itself is in that axis), we 

 know that the illuminating pencil of parallel rays must 

 (in the case supposed) form an angle of 6o° with said 



axis. Having a sheet of cardboard ruled with the 

 principal angles of a semi-circle (as in Fig. 167), 

 instead of the tray before mentioned, we could place 

 the microscope with its tube accurately over the line 

 AC, and the centre of the mirror perpendicularly over 

 the point C. The lamp could then be placed directly 

 over some point in the line marked 6o°, and be raised 

 upon its pillar until a level line from the centre of 

 the mirror passed just above the wick. The bull's- 

 eye would lastly be placed in position, and our 

 intention would have been carried out. 



Now suppose the tube to be sloped to form an 

 angle of 45 with the surface of the cardboard ; its 

 position remaining over the line AC, the centre of 

 the mirror being again brought over the point c, and 

 the lamp (lowered to the altered level of the mirror) 

 remaining on the line marked 6o°. 



The incident pencil will no longer form an angle 

 of 6o° with the axis of the microscope, in agree- 

 ment with the horizontal angle marked upon the 

 cardboard, but one of about 69J . The angle of 

 incidence upon the mirror will therefore be more than 

 34 \° instead of the 30 we proposed to employ, and the 

 mirror, as we have set it, will no longer bring the 

 rays to a correct focus on the object. To get the 

 desired angle of incidence, the lamp must be moved 

 from the line 6o° to that representing the horizontal 

 angle of 45°. 



Now for the reason. The reader must imagine 

 three straight lines meeting in a point at the centre 

 of the mirror : 



(1) one passing up the centre of the tube, known 

 to us already as " the axis " of the instrument, which 

 call A ; 



(2) one passing along horizontally beneath the 

 tube, which call B ; and 



(3) one passing horizontally to the lamp-flame, 

 which call C. 



A and B together form a plane angle (a) in the 

 vertical plane passing through the axis. B and c 

 form a plane angle (/') horizontally, that is, in a hori- 

 zontal plane at right-angles to that of A and B, and 

 on the left side of it ; C and A together form a plane 

 angle {c) obliquely, in a plane sloping upwards from 

 c to A. These three plane angles meet at the centre 

 of the mirror, and form there a solid angle (in shape 

 like a ploughshare) of which they are the "sides." 



It is evident that the oblique angle (c) must be 

 larger than the horizontal one (/>) ; but in order to 

 decide ,how much larger, we need to discover the 

 relations existing between the "sides" of the solid 

 angle. They are the same as those of a right- 

 angled spherical triangle. A simple formula borrowed 

 from Spherical Trigonometry tells us that in such a 



case Cos b = £- — , in which formula a represents 

 Cos a 



the selected angle of slope, c, the double of the selected 



angle of incidence, and b, the corresponding horizontal 



angle which we desire to line off upon the tray. 



N 2 



