of Problems on the Diffraction of Light. 85 



Suppose, in the first place, that each term of the original 

 series has an absolute value which is greater than the 

 arithmetical mean between the preceding and following 

 terms. 



From the forms (2) and (3) we see at once that then 



m 2 m n _ x Q m 1 m n 



If m-L is very nearly equal to m 2 , and m n nearly equal to 

 m n _i, the two limits lie close together, and we may approxi- 

 mately write 



g= m l+ m» (4) 



The last term has been taken as positive for the sake of 

 simplicity. It occurs in (4) of course with its proper sign. 



If the series is such that the numerical value of each term 

 is less than the arithmetical mean of the two terms between 

 which it lies, the same conclusion follows in the same way. 



If in the first p terms of the series each term has a greater 

 value, and in the remaining part a smaller value than the 

 arithmetical mean of the preceding and following term, we 

 may break up the series into two, and obtain the sum 



a m x ,m P m p+ i m n 



It is thus clear that the expressions (4) will be the correct sum- 

 mation only, if the series can be broken up into a small number 

 of separate series for each of which the value of a term is 

 either smaller or greater than the arithmetical mean of the 

 terms between which it stands, so that the sum of all such 

 values as m p —m p+ i may be neglected. In other words, if the 

 absolute values of the terms are plotted as ordinates to equi- 

 distant abscissa, the curve must be either wholly convex or 

 wholly concave, or the number of times the curve changes 

 from concavity to convexity must be negligible compared to 

 the whole number of terms in the series. 



A few remarks are suggested by historical cousiderations. 

 I do not know who first made use of the term " Huyghens 

 zones," but the expression does not seem to me to be 

 altogether appropriate. Huyghens no doubt first divided a 

 wave-front into parts, and considered the effect of the whole 

 wave to be the same as that of the sum of its parts ; but the 

 importance of the so-called " Huyghens zone V lies not in the 

 possibility of division but in the particular manner of dividing 

 into elements, by means of radii vectors differing by half a 

 wave-length. This, as far as I know, is exclusively due to 

 Fresnel, who by its means was enabled to draw important 



