252 Mr. W. H. Walenn on Division-Remainders in Arithmetic. 



direction. Taking the first and repre- 

 sentative square (which has only ones 

 for the numbers in its periphery), the 

 diagonal and the triangular half which 

 has its apex in the lower right-hand 

 corner, its hypotenuse being the dia- 

 gonal, has 11 for all its contained values. 

 As U 7 / behaves itself similarly to U n / 

 in this respect, it is probable that prime 

 numbers for the value of 8 favour this 

 formation. The unitation square of U 6 / is in the margin. 



Of the eight perfect numbers that are known (calling the 

 series p), a few of the unitation functions are : — 



1111111 



12 3 4 5 6 1 



13 6 4 3 3 4 



14 4 2 5 2 6 



15 3 5 4 6 6 



16 3 2 6 6 6 



114 6 6 6 6 



U 2 j>=2, 2, 2, 2, 2, 2, 2, 2. 

 U 3i >=3, 1, 1, 1, 1, 1, 1, 1. 



U 4 .p = 2,4,4,4,4,4,4,4. 

 U 6 p= 1,3,1, 3, 1,1,3,3. 

 L> = 6,4,4,4,4,4,4,4. 



U 7i > =6, 7, 6, 1, 1, 6, 1, 7. 

 U 8 p =6, 4, 8, 8, 8, 8, 8, 8. 

 JJ s p = 6, 1,1, 1,1, 1,1,1. 



U 10 p = 6, 8, 6, 8, 6, 6, 8, 8. 

 U n p = 6, 6, 1,10,6,10,4, 6. 



Algebraical geometry applied to unitation yields some 

 striking results, especially when polar coordinates are used. 

 The polygons obtained by thus setting out the values of func- 

 tions of Us#i in reference to the corresponding and consecu- 

 tive whole-number values of U$# itself, represent graphically 

 those functions in a convenient form. 



The values of functions as given above, also the properties 

 of unitates that are put forth, must be regarded as results 

 given as simple statements ; the proofs of these and of some 

 of the applications of the general formula, together with Tables 

 and methods of calculating unitates, may be found in a series 

 of articles now publishing in the Philosophical Magazine. The 

 published papers are dated : — I. November 1868 ; II. July 

 1873; III. May 1875; IY. August 1875; V. December 

 (Supplement) 1875 ; VI. Jane (Supplement), 1876. 



Unitates manifest themselves as functions which may be 

 found throughout the whole domain of quantity ; they are 

 in some instances interpretable when the quantity from which 

 they are derived is but imperfectly known. 



74 Brecknock lioad, N. 

 August, 1870. 



