io8 PETER GUTHRIE TAIT 



suggested to him a mode of attack on knots quite distinct from that developed 

 by Tait. Taking advantage of Kirkman's extension of the census to knots 

 of eight-fold and nine-fold knottiness, Tait was able to give in his second 

 paper (1884) all the forms of knots of the first seven orders of knottiness, 

 the numbers being as follows : 



Order of knottiness 34567 8 9 



Number of forms i i 2 4 8 21 47 



A year later in his third paper Tait, basing his enumeration on Kirkman's 

 polyhedral method of taking the census, figured the 1 23 different forms 

 of ten-fold knottiness. Higher orders have been treated by Kirkman and 

 Little (Trans. R. S. E. Vols. xxxn, xxxv, xxxvi, xxxix). 



In his second paper Tait pointed out that with the first seven orders 

 of knottiness we have forms enough to supply all the elements with appropriate 

 vortex atoms. 



A curious problem in arrangements suggested by the investigations in 

 the properties of knots was thus enunciated by Tait : 



" A Schoolmaster went mad, and amused himself by arranging the boys. He 

 turned the dux boy down one place, the new dux two places, the next three, and 

 so on until every boy's place had been altered at least once. Then he began again, 

 and so on ; till, after 306 turnings down all the boys got back to their original 

 places. This disgusted him, and he kicked one boy out. Then he was amazed to 

 find that he had to operate 1120 times before all got back to their original places. 

 How many boys were in the class?" 



The answer is 18 (see Proc. R. S. E. Jan. 5, 1880; Sci. Pap. Vol. i, 

 p. 402). 



In his discussion of knots Tait established a new vocabulary and gave 

 precise meanings to such terms as knottiness, beknottedness, plait, link, 

 lock, etc. He introduced with effect the old Scottish word "flype" which 

 has no equivalent in southern English speech, the nearest being " turn-out- 

 side-in." Clerk Maxwell has described some of Tail's processes in the 

 following rhymes : 



(CATS) CRADLE SONG. 

 By a Babe in Knots. 



Peter the Repeater 



Platted round a platter 

 Slips of silvered paper 



Basting them with batter. 



