120 Proceedings of the Royal Irish Academy. 



If an aperture is arranged to show two consecutive teeth, the series 



«>2 + 44 + . . . + «2m+l«"l 



is illustrated. If two apertures separated by the width of a single 

 tooth are employed, the series ^Vs+2 is illustrated, and so on. If the 

 aperture is made large enough to expose four consecutive teeth, the 

 series S^^+i^+aVs is typified. And if there are two apertures, each 

 exposing two consecutive teeth, while u-2 teeth are concealed between 

 these pairs, the arrangement corresponds to the series ^Vi*i%t**rtirt-i' 

 The rest of the screen conceals 



2m +1-(m-2)-4 = 2m - I - u teeth, 



and if this number is greater than m - 2, m is less than m. 

 Take now the triple series 



and suppose 



s < t <u, or t =^ s + X, u = s + X + ^, 



where x and y are positive integers. The series is obviously cyclical, 

 as indeed are all the series in Psm+u so it is sufficient to consider the 

 double series obtained by putting s = 1 ; each term in this double 

 series is the " source " of a single cyclical series which may be written 

 down. The double series is 



The screen must now have three apertures, exposing in general 

 three pairs of consecutive teeth — 



1 and 2, x and x + 1, x + ^ and x + y + 1; 



that is, a pair, x - 2 blanks ; a pair, y - 2 blanks ; a pair, and 



2m+l-2-(^-2)-2-(y-2)-2 = 2m-l-^-y = z-2 blanks. 



In order to find the various arrangements, it is only necessary to con- 

 sider the integral solutions of 



X + ^ + z = 2m + 1 , for which z^ x, and z"^ y. 



When X, y, and z are as nearly equal as possible, z must be the greatest 

 third of 2m + 1, and this is the least value of z. Interchanging given 

 values of x and y of course changes the arrangement. 



