On the Taylorian Theorem. 75 
Hence 
u= Aa’ + A'e' + A”2' uw 
Aya? + Asx og Ng pay. Ce 
B" xa + Bax’ + Ba'a" 
Ax + Aja? + A,r... 
Foeernen 8) 
This being successively differentiated for each of the variables, gives 
dt ng + Bae + = Sits di 
dx x 
= Aw {A+2A,¢+B'r'+B'a"+3A,27 ..... } 
+ Ax {A'+2A,’c'+B’e+Ba"43A,'x'2 .... 3 
+ Ax’ {A"4+2A,"c"4+Ba'+Bia+3A,"2"2...4 
§ d2u Ax? du =Ax® Pe dus Aa’? , a 
dx? 1.2 oie | ose Hast 
At fA, +t SAg ty» wir. giriaiiere 
ee CR eae Bh ee ks t> F 
TED RSE Se) Wee a a ar te 
eee OREM AE Pete Dt ca teiny 
feu) BEAL deus AT Aa ss Na) Ie Ae 7 
dxda 1 dz dz” 1 dat dah Ns 
= Ag. Ag {B’....... } 
ee RB. Tae} 
ee eae 1 Ree ae } 
. . ° . . (E) 
The sum of the series (D) and (2) may evidently be expressed 
thus, 
d?u f Ax ra Ax 12 
eda da’ meh ms 
