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22 THE PHOTOGRAPHIC NEWS. [JANUARY 12, 1883. two concave gratings and one flat. When I tried them I was perfectly amazed. With the concave gratings, nothing is re quired but a slit and a sensitive plate, or an eyepiece if visual observations are required ; the concave surface gives a focus without the aid of any lens. The patience required by my plan now is reduced to ordinary patience, and is less than that re quired for working with a lens. But this is not everything. An easy means of focussing does not mem of necessity good definition ; far from it. Well, I can on’y siy that the definition is equal to the ease of focussing. I will show you a few photo graphs, some by Professor Rowland and others by myself, in 'which are lines that when seen with an ordinarv emating ar of light, which is greater in this grating than in the other, form ing the first spectrum, the 6X2 being the dimensions of the ruled surface. Now the length of the third spectrum with the old grating corresponds with the length of the first spectrum of the new grating, making allowance for the different focal lengths. single, but when seen with this grating are each resolved into two lines, are split up into two. Now the ordinary length of camera and collimator I use for my old grating is 20 inches for each. With this large grating, which is very nearly 6 inches long and 2 inches high, the plate has to be placed about 12 feet from it to get a direct image of the slit in the line of the axis. Using the same width of slit as in the previous calculations, the beam of light forming the first spectrum may be measured by 6x2x20x1=5 144 6 18 where 588 represents the ratio of the distances of the slit from the gratings in the two cases, and 1 the brightness of the beam In the one case the brightness of the white light forming it is certainly not more than 186, and in the other 2% ; so, to get the same length of spectrum, the concave grating has at least seven times as much light, and in fact practically gives a spec trum twice as bright as the second spectrum of the old grating, with which I have taken a great many photographs. It is, how ever, only half as bright as the first spectrum of the old grating. Nevertheless, we have a decided gain by its use when good dis persion is required. I have purposely put the concave grating at it lowest value, and the old grating at its highest. For definition as I have said, the new grating is far superior to the old. I have no doubt that, when we have any sun, I shall be able to get much more in spectrum photography than I have hitherto. I next propose to show, in two simple ways, how the focus may be found mathematically. P Q is the curved diffraction grating, A being the middle point. Take any point, B, near A, and join A B. Let C be the centre of the circle, of which P Q is an arc. Let it be required to find the focus for a ray coming in the direction A K. Join B C, and make C B D = C A K. Let H and E be the points of intersection, as shown. Draw C C at right angles to A 0. Let A C B = • C A K = «* AC=a Join C E. Then since B C = A C, CAB=CBA= (90° — •) * In diffraction spectra, when any particular wave -length has to be calcu lated, the formula used is— ") = sin G — sin i where n Is the order of the spectrum used, A the wave-length, < the interval between the lines, i the angle of incidence, and d the angle of reflection* In the present case either sin G er sin E A B = (90° —•= e) =900— ( ± 0), 2 2 according as B is opposite or on the same side as E. sin 909 — ( • + 0) cos (• + 0) Now A E = A B • 2 = A B 2 sin d sin d C B D = e by hypothesis.