216 Mr. W. H. Walenn on Unitation. 



efficients of the formula are represented in the underneath row 

 as follows : — 



7 4 5! 

 a z a 2 a x I a* 





6 2 5 



Thus written, each ordinal suffix has due relation to the posi- 

 tion of its corresponding digit, and the whole number, in 

 respect to the suffixes, is read: — 5 is the 1st digit (from the 

 decimal point understood), 4 is the 2nd digit, 7 is the 3rd 

 digit. Then (in the opposite direction) 6 is in the first de- 

 cimal place, 2 is in the 2nd, 5 is in the 3rd. 



Normally considered, the operation of unitation always pro- 

 ceeds from right to left ; but the negative suffix indicates the 

 possibility of a change from right to left to left to right, under 

 certain circumstances. This view will receive further consi- 

 deration in the proper place. The use of a Q as determining 

 the place from which the direction of the operation is to be 

 reversed, is believed to be new, and may be of use in other 

 departments of mathematical science. Thus the meaning of 

 a , in the series a d , a 2 , a 1} a , a_ I? «_ 2 , &c, is satisfactorily 

 made out, according to the principles of the interpretation of 

 symbols, to mean the place from which the order of the suf- 

 fixes is reckoned, in reference to direction of counting. 



30. This interpretation of a may be well illustrated by a 

 geometrical diagram : — If a vertical line cross a horizontal line, 

 as in the marginal dia- 

 gram, in the style of Des- 

 cartes's rectangular coor- ___«__ 

 dinates, and if the origin a n • • • "3 a 2 a i a o a -\ a -z • • • a -n 

 be taken as the point from 



which the counting is to 



commence in both horizontal directions, namely backwards 

 and forwards, it is evident that the a which is at the origin is 

 a Q , and that the as with the positive suffixes are distant from 

 the origin in the numerical order of their suffixes; also the as 

 with the negative suffixes are distant from the origin in the 

 numerical order of their suffixes, the negative signs simply 

 indicating that the counting of the suffixes is to proceed in the 

 opposite direction to the counting of the positive suffixes. 



31. The substitution of (r — B) for r, in the formula 



a n r n ~ x + a n .,r n ~ 2 + . . . + a z r 2 + a 2 r + a x 

 for a given number N, yields 



"».(«— «)""' + «»-.(»— 8)"- ! + ...+a 3 (V- Sy + a 2 (r- B) + ai , 



Tliis is a number ^Yflich has the same remainder to 8 as N has ; 



