> 



+ 



+ 



+ 



+ 



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+ 



II 



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+ 



+ 



+ 



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G 



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F 



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E 



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+ 



+ 



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+ 



I) 



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+ 



+ 



+ 



+ 



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C 



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+ 



+ 



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B 



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A 



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K 



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L 



L 



PROFESSOR TAIT ON KNOTS. 341 



Hence, if the following be a portion of the table, the crosses being placed for the various values of 

 p T ,, nil or not, 



Values of r. 

 012345678 







1 



«(-! 2 



o 



03 3 



4 



5 



6 



7 



it will be seen at a glance that the above equation tells us to add the numbers A, B, C, D, E together 

 to find the number at K. This is quite general, so that L, in the second last column, is the sum of 

 A, B, . . . . , H ; and all the numbers beyond it, in the same row, are equal to it. In the table on next 

 page, each number corresponding to the first L is printed in heavier type, and its repetitions are taken 

 for granted. 



Thus it is clear that simple addition will enable us to construct the table, row by row, provided we 

 know the numbers in the first row and those in the first column. Those in the first and second columns 

 are all obviously zero, as above. The rest of the first row consists of units. These are the values of 

 p r , i.e., the first term of the expression above for p T r . Hence we haVe the table on the following page, 

 which is completed only to r = 17, with the corresponding sub-groups. 



Erom the table we see that p\ = 8. Hence the partitions of 18, subject to the conditions, are in 

 number 



8 + 11 + 11 + 14 + 10 + 8 + 3 + 1 = 66, 



which agrees with the detailed list in § 7 above. 



[The rule is to look out the number p" n , and add it to all those which lie in the diagonal line drawn 

 form it downwards towards the left. But the construction of the table shows us that this is the same 

 as to look out p? n at once.] 



Similarly we verify the other numbers of partitions given in the text. 



And it is to be remembered that p£ is the number of required partitions in which n occurs, and that 

 every one of the class p"^. has for its largest constituent n - r. Thus, looking in the table for p] and 

 the numbers in the corresponding downward left-handed diagonal, we find the series 



4 6 5 5 2 1, 



which will be seen at once to represent the dissection of the partitions of 1 4 given above. 



The investigation above was limited by the restriction, imposed by the theory of knots, that no par- 

 tition should be less than 2. But it is obvious that the method of this note is applicable to partitions, 

 whether unrestricted, or with other restrictions than that above. The only difficulty lies in the border- 

 ing of the table of double-entry. Thus, if we wish to include unit partitions, all we have to do is to put 

 unit instead of zero at the place r= 1, s = 0, and develop as before. Or, what will come to the same 

 thing, sum all the columns of the above table downwards from the top, and write each partial sum 

 instead of the last quantity added, putting unit at every place in the second column. 



Similarly, we may easily form the corresponding tables when it is required that the partitions shall 

 be all even, or all odd. 



