474 On Inverse Orthogonalism and Sign-successions, 



The meaning of this Table is self-apparent. Thus, ex.gr. } if we 

 wish to find the value of ( \- + ), i. e. the number of recur - 



+ \ 

 rences of + in the three given series, we read it off from the 



+ 

 5th line above and find it equal to 



S S ] ~f~ Sq~T ^3"" #1, 2~~ $1, 3"T~ $2, 3 $1, 2, 3 



8 



The Table of signs itself is obviously the matrix corresponding 

 to the product 2.2.2. 



From the fact of this Table being orthogonal, we infer that the 

 two sets of quantities are (to a numerical multiplier pres) the same 

 linear functions, the first set of the second, and the second of the 

 first. 



14. The theorem of art. 10 may be extended to simultaneous 

 progressions of signs denoting any root of +, as, ex. gr., +p, p 2 , 

 where p is a cube root of + instead of -f and — . Let each 

 series be supposed to consist of qih roots of -f, and let 

 (Xj, X 2 , . . . , A.) denote the number of recurrences of the column 



\, • 



• 2 in which each \ is some ^th root of +; then there 



will be ^quantities of the form (\„ X 2 , ...,X ? ). Again, we 

 may form series by combining together not merely the given i 

 series themselves, but their squares, cubes, &c. up to the (q— l)th 

 powers, and form the term-to-term products of all the series 

 entering into any such combination ; in this way, including s (the 

 series constituted exclusively of -f signs), we shall obtain q l series, 

 the general symbol for the sum of the terms in any one of w T hich, 

 when we substitute the roots of 1 for the corresponding roots of 

 + , may be written [sji, s| 2 , sf 3 . . . , s?i], where each s is a qth. 

 root of +, and each q with a subscript is some one of the num- 

 bers in the series 0, 1, 2, . . . (q — 1). If now we understand 

 by the above bracket, when q v q 2 , . . . , q { are all zero, the value 

 corresponding to s in the particular case previously considered, 

 i. e. the number of terms in each series, the relation between 

 the two sets of cf numbers is given by the equation 



* The reader will please to observe that the terms included under the 

 sign of summation are in general not real but complex numbers formed 

 with the gth roots of unity. Their sum, however, is necessarily a real 

 number, being the number of recurrences of the columnof signs X J; X . . . ., X 



