AND IMAGINARY EOOTS OF EQUATIONS. 
603 
Let the derivative of this function be taken in regard to v, and we have 
¥'(u,v)= /3^( a 2 u + /3 2 vf + /3 3 j 3 ( a 3 u + . . +[5 n s n (a n u+[3 n v) 2i , 
where i3 2 s 2 , (5 3 e 3 . . . (3 n s n have the same signs as s 2 , s 3 , . . . s n respectively. 
Now the pencil-type of F '(u, v ) will be the per-rotatory type g n , . . . g> 2 , of which 
by construction the variation-index is v. Hence by hypothesis F \u, v ) has not more 
than v real roots, i. e. at least 2 i—v imaginary roots. Hence F (u,v) has at least that 
number of imaginary roots, i. e. at most (2t+l) — (2i— v), i. e. y+1 real roots. Hence if 
the theorem is true for v an even number, it is true for j / + 1 . 
In like manner let us proceed to show that when it is true for v an odd number, it 
would remain true for I' + l. 
The reasoning will be precisely similar to that followed in the antecedent case. We must 
find a phase of the per-rotatory type g>„, g> re _ 15 . . . g> 2 , having the variation-index v such 
that the trans-rotatory reduced type g n , § n _ y , . . . f 2 shall have the variation-index v — 1 ; 
the new pencil will still continue to be a type-pencil of the given superlinear function, 
the change of direction in the bunch of rays one on side of being now unaccompanied 
with change of sign, such change corresponding to z(ax-{-by) 2i becoming changed into 
s {—ax— by) 21 without e undergoing a change of sign. 
As before, the axes of coordinates are transformed from |, tj into ?/, and we obtain 
F (u, v')=e l (u 1 u) 21 -\-e 2 (a, 2 u-}- fi 2 vy‘ s n (oc, n u-{- (^ n v) 2n+1 , 
^ F'O, v)=f3 2 z 2 (a 2 u+(3 2 vf - 1 -\-. . .+p n e n (ci n u-t-f3 n v) 2i . 
for which the type-pencil is the trans-rotatory type g n , . . . § 2 , of which by construction 
the variation-index is v — 1 , so that its number of imaginary roots is 2 i — (v — lj, and con- 
sequently the number of real roots of F (u,v) will be v+1. 
Thus, then, if the theorem be true for v, whether v be even or odd, it will be true for 
*'+!• 
But when v=0, the superlinear function becomes a sum of even powers of linear func- 
tions of x, y, all taken with the same sign, of which the number of roots is evidently 0. 
Hence, being true for this case, the proposition is true universally. 
It will be noticed that the algebraical part (as distinguished from the purely polar- 
tactic part of the above demonstration) depends on the same principle of which such 
abundant use has been made in the former part of this dissertation, viz. that the num- 
ber of imaginary roots in any ordinary algebraical equation in x cannot be increased 
when we operate any homographic substitution upon x, and take the derivative of the 
equation thus transformed in lieu of the original( 24 ). 
( 24 ) For greater clearness I present in an inverted order of arrangement a summary of the foregoing argu- 
ment. 
By an ith derivative of f(x, y) is meant any derived form 
MDCCCLXIV. 
