EQUATIONS OE THE EIEST OBDEE. 
445 
whence, substituting in (9.), 
C?P I I TT J XT \ 
dxp+i du■^ dx^ • • • + pi 
+~(;fr- + 'H.,,p...+B.p,p)=0, 
' dup \dxp+i ' dxi • dxpj ’ 
or 
(“•) 
and in this equation we may give to ^ the successive values 1,2,..,^. If, then, we write 
4^+(A,«.)^^...+(A,«,)4=A'„ 
we see that the proposed transformation will have the effect of converting 
^25 ••• 
into 
^^25 ••• 
respectively, and the system of partial differential equations into 
a;p=o, a;p=o, ... a;p=o. 
The developed form of the first of these equations is 
I /A I /A n 
du^ 
But since Ui, ... Up are integrals of AiP=0, we have 
Ai2^i=0, ... Ai2^p=0, 
so that the equation A',P=0 reduces to 
dF 
dx 
: 0 . 
■p-k-\ 
We learn from this that Wp^^ will not explicitly appear in P after the transformation 
which introduces Wj, ... Up. 
The developed form of the remaining^ — 1 equations represented by (10.) will be 
dx 
. /A , /A A 
... +(A2^g , — 0, 
‘P+2 
}- 
dF , . .f/P , . . 
+ du^ "• dUp~^ ’ 
( 11 .) 
and we shall next show that the variable Xp+■^ will not present itself in the coefficients 
(AjWi), &c. 
The general form of such coefficients is where i has any value from 2 to m, and 
s any value from 1 to p. 
Now if in (5.1, which is true independently of the nature of the function P, we make 
MDCCCLXII. 
3 p 
