42 
The differential theorem may be satisfactorily established 
by conducting the investigation thus : 
Let 
a — ba + cx? — de’? +&e. = 8 
— br + cz? — dx? + &e. = s—a (10) 
 — b+ cx 4 dz? + ex? — &e. ==. (11) 
Consequently, by adding these two equations together, and 
representing the numerical differences b—c, c—d, d—e, &c. 
by A, A’, A”, &c,, there will result the equation 
— b6—A.2+ A.2?— A" 2? + &e. = + : (s—a) (12) 
*, —ba— A.2? + Aa? — A”.2* + &e. = on + 1) (s—a)=s' 
s/ 
et Tse > a; 
that is, 
bx / ” 
dhs TS #4 A.x + A’.a2?—A”.x* + &e.] (13) 
And by treating the series within the brackets as the original 
was treated, and so on, we shall finally obtain the transfor- 
mation 
a+1 (+1)? («+1)° 
or putting a = 0, and dividing by — x, we have 
ch alors fies tg = 
b A’. ire A*. 23 
ei * (erie Ga tGaay TG ee 
which is the usual form of the theorem. 
Now the preceding reasoning is inadmissible except the 
proposed series be convergent; that is, except rx” approaches 
to zero as m approaches to infinity, 72” standing generally for 
the n term of (10). Forin (12), which results from the sum of 
(10) and (11), this x”, or final term, is regarded as zero, and 
is neglected; inasmuch as it is by this term that the series (10) 
EE EE 
