S4 THE ROYAL SOCIETY OF CANADA 
In equation (4.) we now substitute 
tig Vi = Uo + Vo = Q 2°’ +p, 
where for a’ we select one of the roots of the equation of alternatives 
(of degree 7’) associated with equation (4.). This gives us an equation 
n 
1 
18. F, (z, v2) -y— FO (ue) v5 =0 
el 5! 
in which the element of the summation not involving %—4.e. the 
element Fi (z, #)—has an order > c, > ¢,. 
Selecting by the above process the successive elements in a sum 
19. Wy+uet....+u,=az+a'2/+....+a8-) ge ae 
the functions 
B20). F ei), MR be e Pee, tip.) 
that is to say, the functions . 
20. F (20) 7 (6,10), Pi AEs) ns (F (2, Wi+uo+..+ug) 
present a successively increasing sequence of orders c,, c,,.... c®, 
It is readily seen that the total possible number of sums of k 
elements #1+u2+...... +u,, each differing from the others by some 
one element at least in the succession, cannot exceed the amplitude 
of the original equation. If namely o be the number of distinct roots 
of the equation (7.), we have just o alternatives in choosing the 
co-efficient @ in the element u,=a2° and therefore just o alternatives 
in choosing the element 7 itself, since the exponent c is uniquely 
determined. On indicating by fi, po,..... bo respectively the mul- 
tiplicities of the o roots of the equation (7), we have seen that the 
choice of a root a of multiplicity p4 gives rise to an equation for % 
of amplitude 7’, < py. Since pit....+pfo =7 we see that for the am- 
plitudes of the several equations for v, corresponding to all the 
different possible determinations of the first element # in the sum 
(19.), we have 7i+..... +rg<r. That is, the sum of the amplitudes 
of all the equations which give rise to the determination of the 
second element #2 in (19.), after the determination of a first element 
u, is equal to or less than the amplitude of the equation (1.) which 
gives rise to the determination of the first element w. In like 
manner it evidently follows, that the sum of the amplitudes of all the 
equations which give rise to the determination of a third element #3 
in (19.), after all possible determinations of the first two elements 
uw, and ue, is = rit....+%o<r. In like manner it evidently follows, 
that the sum of the amplitudes of all equations which give rise to the 
determination of a (k+1)th element #71, following all possible 
selections of the succession of k elements 2, #2,....ux, is € ? and 
therefore the number of these equations must certainly be < 7 and the 
number of all possible selections of the succession of k elements 
U1, U,....ux Must consequently also be <7. However large k may 
be then the number of sums of the type (19.), constructed by the 
process here in question, can never exceed 7 the amplitude of the 
equation F (5, v) = 0. 
