n8 PETER GUTHRIE TAIT 



circles of curvature, attractions, centrobaric distributions, logarithms, etc. 

 The note on centrobaric distributions he afterwards simplified and extended 

 in his booklet on Newton's Laws of Motion, and gave a remarkably simple 

 geometrical proof that the potential of a uniform spherical shell is constant 

 throughout the interior, and varies for external points inversely as the distance 

 from the centre. 



The last published paper not connected with quaternions was on a 

 generalization of Josephus' problem (1898, Proc, R, S. E. Vol. xxn). The 

 original problem stated simply is to arrange 41 persons in a circle in such 

 a way that when every third person beginning at a particular position is 

 counted out, a certain named one will be left. What position relatively to 

 the first one counted will he occupy ? It is said that by this means Josephus 

 saved his life and that of a companion out of a company who had resolved 

 to kill themselves so as not to fall into the hands of the enemy. Josephus 

 is said to have put himself in the 3131 place and his friend in the i6th place. 

 Tail's generalization consists in pointing out that, if we know the position 

 of "safety" for any one number, we can without going through the labour 

 of the obvious sifting-out process at once say where the position of " safety " 

 will be if the number is increased by one. This position is simply pushed 

 forward by as many places as there are in the grouping by which the successive 

 individuals are picked out. By successive application of the process, Tait 

 quickly found that if every third man is picked out of a ring of 1,771,653 men, 

 the one who is left last is the occupier of place 2 in the original arrangement. 

 Hence if there were 2,000,000 in the circle the place to be assigned to the 

 last one left after the knocking out by threes is evidently 



2 + 3 x (2,000,000 - 1,77 1,653) = 2 + 3 x 228,347 = 2 + 685,041 = 685,043. 



When the number reaches 2,657,479 a new cycle will begin with the 

 place of safety in position i. The general rule given by Tait is: 



" Let n men be arranged in a ring which closes up its ranks as each individual 

 is picked out. Beginning anywhere, go continuously round, picking out each ftith man 

 until r only are left. Let one of these be the man who originally occupied the /th place. 

 Then if we had begun with + i men one of the r left would have been originally 

 the (p + *)th, or (if p + m>n + i) the (/> + -- i)th." 



