AND IMAGINARY ROOTS OF EQUATIONS. 601 
P represents a continual succession of signs of the same name as + + . . . + or ... — , 
and O represents a succession beginning with one sign as + and ending with one or 
more signs — , or else beginning with — and ending with a succession of signs + . 
Essentially, then, as a change of signs throughout a whole succession does not affect 
the variation-index, we may suppose 
P=+ + +, 
the signs intervening between the two expressed signs — in being filled up in any 
manner whatever, and those between the two signs + with signs exclusively + . 
Let now that phase of O be taken which commences with the first sign of the final 
succession of + . Then O becomes 
(<&)=+...++....+, 
which is of the form 
+ .....++, 
so that P is only a particular case of (O). If the last sign in (ff>) be suppressed and 
the result treated as a per-rotatory type be called (<3>)', so that (+)' = + + , we 
have variation-index in (+) — variation-index in (T>)'= changes of sign in \- less 
changes of sign in + + = 1 — 0 = 1. 
Hence the proposition is established for both cases. 
(17) The theorem to which this Lemma-proposition is to be applied concerns equa- 
tions of the form 
gi< +s 2 < + 0 . . . +g„C=0, 
where w 15 u 2 , . . ., u n are any linear functions of x, y, m is any positive integer, and 
g 1? g 2 , . . . s n are each respectively and separately, either plus unity or minus unity. 
Such an equation for convenience of reference may be termed a superlinear equation, 
and the function equated to zero a superlinear function. 
Every superlinear function may be conceived as having attached to it a pencil of rays 
constructed in a manner about to be explained. 
1. We may conceive the function to be prepared in such a manner, that supposing 
ax-\-by to be any one of the n linear elements u, every b shall be positive. If m is even, 
this can be effected by writing when required for ax + by, —ax— by without further 
change. If m is odd, we may write when required —ax— by in place of ax -{-by, 
changing at the same time the factor g, which appertains to ( ax—by) m from +1 to —1, 
or vice versa , from — 1 to +1. 
Now take in a plane any two axes of coordinates 0|, Of], and consider a, b as the £ 
and f] coordinates of a point. All the n points thus obtained, on account of every b being 
positive, will lie on the same side of the axis Or,, and thus the entire n linear functions 
will be represented by a pencil of n rays, the two extreme rays of which make an angle 
less than two right angles with each other ; but each term of the superlinear function 
contains, besides ( ax-\-by) n , a definite multiple +1, or —1, and we must accordingly, to 
