and Simultaneous Sign-successions. 473 



Genocchi has followed me in regarding the theorem as an in- 

 dependent one, and devoted an article to the demonstration of it 

 as such in the Nouvelles Annates for January of this year*. 



13. The complete system of relations between the two sets of 

 2* quantities given by the theorem in art. 10 may it is evident 

 be expressed by means of the inverse orthogonal matrix (also 

 orthogonal) whose type corresponds to 2 . 2 . 2 . . . (i terms). 

 Thus, ex. gr. } for the case of i — 3, we may write — 



2 3 .(+ + +) 

 2 3 .(+ + -) 

 2 3 .(+-+) 



2M+ ) 



2 3 .(-++) 

 2M- + -) 



2M +) 



2M— -) 



s 



s l 



5 2 



*3 



s h 2 



S l, 3 



S 2, 3 



S h 2, 3 



+ 



+ 



+ 



+ 



+ 



+ 



+ 



+ 



+ 



+ 



+ 



— 



+ 



— 



— 



— 



+ 



+ 



— 



+ 



— 



-U 



— 



— 



+ 



+ 



— 



— 



— 



— 



+ 



+ 



+ 



— 



+ 



+ 



— 



— 



+ 



— 



+ 



— 



+ 



— 



— 



+ 



— 



+ 



+ 



— 



— 



+ 



+ 



— 



— 



+ 



+ 



— 



— 



— 



+ 



+ 



+ 



— 



* If we call v the number of real roots in/ comprised between a and b, 

 we know from Fourier's theorem that p=Ap — 2d, where 6 is the number 

 of times that an even change occurs in the value of the p as we pass from a 

 to b, this change being always in the positive direction. And, again, as I 

 have shown in the article in the Philosophical Magazine above referred to, 



_ A/J + A0 



P = 



&> 



where S is the total number of times that <p undergoes a change within the 

 same interval,— such change being always even, on account of the two ter- 

 minals of the G series being both positive — the one extremity being a positive 

 constant, and the other the square of/. This change, however, is sometimes 

 additive and sometimes ablative, cp not necessarily increasing always (as p 

 does) on ascending from a to b : thus the two unknown transcendants 6 

 and S are connected by the simple relation 



20-$= A ^- A( ft. 

 2 



Of course each evanescence of a term in the / or G series between two 

 terms of like sign is to be reckoned as a distinct time of change. I also 

 make abstraction of the singular cases where several consecutive terms va- 

 nish together in either series. 



Phil. Mag. S. 4. Vol. 34. No. 232. Dec. 1867. 2 I 



