470 Mr. J. J. Sylvester on Inverse Orthogonalism 



the original series of fs and G's. Now it appeared to me desi- 

 rable, in the same way as double and higher orders of denume- 

 rants have been shown in my lectures on Partitions of Numbers to 

 be expressible as linear functions of simple denumerants, so in like 

 manner to get rid^of compound variations and permanencies, and 

 to express them, or at least their number, by means of simple va- 

 riations or permanencies. This comes to the same thing as find- 

 ing a means of making the enumeration of the four species of 



+ — 



in two simultaneous series of 



+ 



+ 



compound signs 



signs, depend on the enumeration of the simple signs + or — 

 in those series themselves, or in series derived from them, or in 

 the two sorts combined. 



10. As a first step in the generalization of this question, let 

 us suppose i series of simultaneous progressions of + and — 

 signs giving rise to 2 l varieties of vertical combinations of sign. 

 Now let the i given series be combined, r and r together, in 

 every possible manner, where r takes all values from to i\ both 

 inclusive. 



When r=l, it is of course understood that the so-called com- 

 binations are the original i series themselves. 



When r = 0, it is to be understood that a series exclusively of 

 the signs + is intended. 



When r is not 0, nor 1, let the r series corresponding to my 

 r-ary combination be multiplied term-to-term together. 



When r = 0, the -f- succession, and when r=\ the given n 

 series are to be reckoned as the corresponding products. The 

 number of series of signs so obtained will of course be 



1+ i + i|i=i) + .,,. =s ,-. 



By the sum of any series let us understand the number of signs 

 4- less the number of signs — . When the i given series are 

 written over one another, each of the 2 £ varieties of columns 

 that can be formed of the signs + and — will occur a certain 

 number of times. I shall show that these 2' numbers are linear 

 functions of the 2* sums last mentioned. Of this theorem, on 

 account of its importance, I shall give a rigorous proof. 



As a matter of typographical convenience, I write the columnar 

 combinations of sign in horizontal in lieu of their proper vertical 



+ 

 order, as, ex. gr. f -f + — in lieu of + , and, moreover, use such 



horizontal line enclosed within brackets to signify the number 

 of the recurrences of the corresponding combination; thus 



