82 THE ROYAL SOCIETY OF CANADA 
4. Fi (z, %1) = SF) v5 = 0, 
where for brevity we write F(u,) for F9 (zg, 1). 
1 
Co —Cs 
Let 7 be the greatest value of the suffixes s for which 
We then have 
5. 
Getta SiGe Goes) Sit 
CCE Cr OS 
CCS CIC ST 
The orders of the coefficients = F) (uw) we shall indicate by the 
letters c’;. Let us first consider the order c’, of 
n 
6. F (zg, m) = Yat xs + f, (2). 
= 
Here it is evident from (5.) that no element in the sum is of order 
lower than c,. Furthermore from (5.) it is also evident, that in order 
to give F (z, 4) an order which is >co, it is necessary and sufficient 
to choose the as yet undetermined number a so as to satisfy the 
equation 
re f (a) =f, (0) a’ +....+fo(0) =o, 
where a power a’ has as coefficient o or f, (0) according as c; + s¢>C, or 
=C,. This equation is of degree 7 precisely. Choosing a root of the 
equation for a we have c’, >c. The number of the alternatives open 
to us in choosing a is not greater than 7. This number 7 is a perfectly 
definite number connected with the equation (1.) and having reference 
to the value z=o. For temporary convenience and for lack of a 
better term we shall call it the amplitude of the equation (1.) for the 
value 2=0. The number of the alternatives open to us in choosing 
the coefficient a in an element “,=a 2°—where c is determined by the 
formula (3)—so that the order of the term not involving % in 
equation (4.) may be greater than the order of the term not involving 
v in equation (1.)—is therefore never greater than the amplitude of 
the equation (1.) It is in fact just equal to the number of distinct 
roots which are possessed by the equation (7.) whose degree coincides 
with the amplitude of the equation. The equation f (a)=o we shall 
call the equation of alternatives belonging to the equation F(z, v) =o. 
It evidently does not have 0 as a root. 
Now assume a to be a p-fold root of the equation (7.) and let us 
consider the order c’; of 
n n 
t! 
8 F) (1) => TE MtTt= D pe D EE an 
t=s 
t—s/ 
t=s 
Here from (5.) we have & +(t—s) c>c —sc for t>r and conse- 
quently c’,>c,—scfors>r. At the same time we have c, = Co —7C 
and therefore c’, =cg—re. Fort<rwehave © + (t—s)c¢ > c —sceand 
as a consequence ¢’,>¢o —sc for s<r. Supplementing (5) we then 
have 
