and Simultaneous Sign- successions. 471 



(H 4-) means the number of times the combination 



Q 



occurs in four given simultaneous progressions. Again, as re- 

 gards the sums, S will denote the sum of the line of plus signs, 

 which is of course the same as the number of terms in each pro- 

 gression, i the number of columns, and S p> ?j r . . . . will denote the 

 sum of the line formed by the multiplication together of the pth, 

 qth, rth, .... lines of the given i set of lines. This being pre- 

 mised, and using each of the symbols X l5 X 2 , X 3 , .... to denote 

 + or — , as the case may be, the number of recurrences of each 

 species of combination in terms of the sums is expressed by the 

 following formula, 



[Xj, X 2 , . . . X-] 



as I shall proceed to prove. But first, to make the meaning of 

 this formula more clear, let us suppose z = 2, the formula then 

 gives the following equations : — 



(+ +) i. e. the number ofl f+1 i < > 



combinations / [_ + J =? ^ S+ ° l + * 2 + Sl ' 2 * ' 



(+— ). » » [_J =i{ S + S l ~ S 2~ S h2\, 



(— .+ ) » » [ + J =J{«- »i + *a-*i,a}, 



(--) » » [l]=i{ 5 - 5 i- 5 2 + 5 i, 2 }- 



11. Now for the proof of the general formula. 



For shortness call the quantity S + S\ p s p + X\p\ q S P; q . . . 

 (where the signs X 2 , X 2 , . . . X; are all supposed to be given) E. 



Let us consider the effect of the existence of any single column 

 of signs jju v /x 2 , . . . , fit in the given i progressions upon the value 

 of E ; besides contributing the signs fi lt /z 2 , . . . , /-t; respectively to 

 the series s v s 2 , . . . s { this column will contribute to the series 



se lt e 2 , . . . o f , the sign fj, 01 , /*<,,, . . . , fi ej . 



Hence altogether it will contribute to E 



(1 +X l/ Lt 1 )(l + X 2 /z 2 ) ... (1 + \ i /n l ) units ; 



and thus the total value of E, depending on the entire number 



