618 TRANSACTIONS OF SECTION A. 



If a particular solution of the equation Xy = .f" is required it is furnished by 

 (}){m, .v)IFQ{m) unless Fj,(m) is zero, and when Fo(?«) = by 



It i ^"'^'"> 

 and similarly in other cases. 



It is also possible to meet the' difficulty that arises when Fy(m) = has two 

 roots differing by a multiple of r, say fi and fi. + br, where 6 is a positive integer. 

 Take A^rzin — fx instead of 1, and Aj, Aj . . . A,,_i are all algebraic fractions in m, 

 each having ni — p. as a factor in its numerator. A^, A^j+j . . . will not in general 

 have this factor, since it occurs in Y^{m + br), the coefficient of At, in the equation 



Ai,Fo(m + 6r) + Aj_,Fi(w + Zp/--/-)+ ... =0 



by which Aj, is given. We now have 



X^(m, .r) = F(,(m) {m - /i).t."' 

 80 that the factor m — ji. is repeated on the right, and when m =/i 



rf) and rs - 

 dm 



are both solutions of Xy = ; differs only by a constant factor from <f){fj, + br, x), 

 but d<f)ldm is an independent solution. 



In special cases A;,, Ai,+ j . . . may have the factor m — fi, and then (^(^, .7) 

 vanishes identically, d(pldfi is a series of the ordinary type without logarithmic 

 terms. 



If there are three roots differing by multiples of r, let them be ;i, fi + br, fi + cr, 

 where b, c are both positive or zero. Then take Ay = (w — ju)'-, and the three 

 solutions corresponding are 



, d<f> d'4> , . 

 cm cm- 



The functions involved are such that the change of order of the differentiations 

 can be easily justified. , 



Department op General Physics. 



The following Papers were read :^— * 



1. A New Three-Colour Camera. 

 ByS,iv W. DE W. Abney, K.C.B., F.K.S. 



Some two years ago I brought out a new form of ' one-exposure ' camera for 

 three-colour work. The problem was to bring three images of the same object 

 into focus and side by side on to one photographic plate, and also to make the 

 angle of the included image of not less than 30°, and to impress all three images 

 with one exposure. Of course if we have a wide camera and place three lenses 

 equidistant from one another, and of exactly the same form, we obtain what we 

 want, for we can expose them simultaneously. If we require a quarter plate 

 image the lenses would have to be 3^ inches apart, with the result that when we 

 come to print from the three negatives and superpose the prints one over the 

 other, as in the Sanger-Shepherd lantern-slide process, or in the ordinary typo- 

 graphical block process, we shall find that, owing to what may be called the 

 stereoscopic effect produced by the distance apart of the lenses, the images 

 will not fit. Two years ago I published the way by which the stereoscopic 

 effect could be reduced to a minimum, and it became practically non-existent. 

 Baldly it may be said that three simple lenses of very narrow aperture, about 



