532 Mr. J. W. L. Grlaisher on Applications of 



Thus taking, for example, the formulas of § 6 in the case of 

 r=l 9 we have 



<1) = 1 



<r(2)-o-(l) = 2 



(7(3) -<r(l) = 3 



<r(4) -o-(2) =4 



<r(5) -o-(l) = 5 



o-(6) -o-(3)-o-(2) + o-(l) = 6 



and 



<r(l) = l 



o-(2)-2o-(l) = l 



o-(3) - 3(7(1) = 1 



o-(4) -2(7(2) =1 



(7(5) -5(7(1) = 1 



<7(6) - 2(7(3) - 3(7(2) + 6(7(1) = 1 



«r(n) = (-)' 



From the first system of equations we find: — 



oo 



1, 2, 3, 4, 5, 6,.. 

 1,-1,-1, 0,-1, 1,.. 

 0, 1, 0,-1, 0,-1,.. 

 0, 0, 1, 0, 0,-1,.. 

 0, 0, 0, 1, 0, 0,.. 

 0, 0, 0, 0, 1, 0,.. 



And from the second system: — 



1, 1, 1, 1, 1, 1,.. 

 1,-2,-3, 0,-5, 6,.. 

 0, 1, 0,-2, 0,-3,.. 

 0, 0, 1, 0, 0,-2,.. 

 0, 0, 0, 1, 0, 0,.. 

 0, 0, 0, 0, 1, 0,.. 



In the first determinant the first row consists of the 

 natural numbers ; the second row is formed by entering 1 



