Mr. W. H. Walenn on Unitation. 549 



The series for performing the division to obtain the period 

 by means of unitation (art. 21) is that of Ui 7ft=7, 4, 1, 8, 

 5, 2, 9, 6, 3, 10. 



23. Similarly for reciprocals of the form -^ tj> Since 



Yj_ l =3. 1Q , these reciprocals have 3 for the last figure 



u io o Ujo y 



of their period. The multipliers to construct these reciprocals 

 from right to left are comprehended in the arithmetical series 

 1, 4, 7, 10, &c, the corresponding denominators being 3, 13, 

 23, 33, &c. In this case the function U 10 ^ yields the figures 

 of the quotient by multiplication by 3, and the series for per- 

 forming the division inversely is that of U 10 3ft = 3, 6, 9, 2, 5, 

 8, 1, 4, 7, 10. 

 Also by similar processes to the above, reciprocals of the 



brm ztk ^ are found to have 9 for the last figure of their 



10ft — 9 



period. The multipliers to construct them from right to left 

 are comprehended in the arithmetical series 10, 19, 28, 37, &c, 

 the corresponding denominators being 11, 21, 31, 41, &c. The 

 function Uio r yields the figures of the quotient by multiplica- 

 tion by 9 ; and the series for performing the division inversely 

 is that of U 10 Hn=l, 2, 3, 4, 5, 6, 7, 8, 9, 10. 



Ill 



The other forms of reciprocals are 





10/i -2' lOn -4' lOn -5' 



? and — ^ 5 • When reduced to decimals, these are 



LOn— 6 10ft— o 



either finite, or they can be derived from the form .. — -: by 

 multiplication or division. For instance, -^\ = '015625, and 



i_ i _ 7 JL -^1 JL. L-Ji— 1 . 1 . 



is Unite; 54- 2727"" 2 ' 189 ~ 2 ' 189 ' 72 ~ 8.9 " 8 9 



All other decimal equivalents that are derived from fractions, 

 being multiples of reciprocals, can be obtained by multiplica- 

 tion in connexion with one of the above methods. 



The employment of unitation to investigate the constitution 

 md properties of decimal equivalents derived from fractions 

 shows the analytical power of unitation, and points out another 

 method of checking calculations into which recurring decimals 

 mter. 



74 Brecknock Road, N., 

 April 1876. 



[To be continued,] 



