Mr. W. H. Walenn on Unitation. 215 



pression ; this is the case in the formula 



sin (~x D^X = A^- A 3 A' 3 + A 5 a?- A 7 # 7 + "&c, 

 derived from 

 sin C^xD\X = sin (| . (A A -f sh/|" . 1^ A^ + sin (~ . 2 J A 2 x 2 + &c. ! 



But no full recognition of the function of a suffix appears in 

 any of these uses. When this method of indicating the order 

 of a series of quantities is completely developed, it should be 

 capable of showing, not only the sequence and direction, but 

 also the relative position of each member of the series to a 

 given point, as in the case of the ordinal numbers, 1st, 2nd, 

 3rd, &c. 



The use made of the suffix in unitation, as proposed in 

 art. 28, is in accordance with these views ; and, in the general 

 formula for integer numbers above cited, the suffix expresses 

 the order of the digits commencing with the unit's digit, count- 

 ing the unit's digit as the 1st integer digit from the decimal 

 point, the tens' digit as the 2nd digit, &c. A question then 

 arises which is as important, in relation to ordinal numbers, as 

 the meaning of a is in the theory of exponents. This question 

 can be answered on a basis as logical as that of the exponen- 

 tial question : it is, " If the series be continued towards the 

 right hand, as 



a n> a n-\) a n—2) a n-3) & c « • • • a 2i a l) a Q) a -\) a -2) &C, 



what does a mean?" It is impossible to think of the next 

 number to the right of the unit's digit (which is in the 1st 

 decimal place) as having any relation to a ; for as the unit's 

 digit in the number is the 1st digit to the left of the decimal 

 point, so the next figure to the unit's digit, on the right hand, 

 is the 1st digit to the right of the decimal point. If these 

 suffixes are to be read as ordinal numbers, a must be left out ; 

 for the Oth place to the right or left of the decimal point is the 

 decimal point itself. That is, if the above general formula be 

 extended to decimals (or towards the right hand), it must be 

 written 



a n r n ~ l + a n ^r n - 2 + ...+ a- 6 r 2 + a 2 r l + a x r° + a^r~ l + a^ 2 r~ 2 + ... 



and a has no other meaning than the decimal point itself. 



Writing in the top row, which follows, an ordinary number 

 with a finite number of decimal places, the corresponding co- 



* See Dr. Graves (Bishop of Limerick), ' Law's Mathematical Prize,' 

 1833, quoted in Carmichael's 'Calculus of Operations/ p. 1(0. 



