EQUATION'S OE THE EIEST OEDEE. 
447 
when ^p+i=:0 each Wy shall reduce to Xj. For this purpose it is only necessary to choose 
as arbitrary constants 'a set of arbitrary values of X 2 , ... Xp corresponding to ^^^. 1 = 0 . 
Let x\, x^^ ...Xp be such arbitrary constants, and let the given system of integrals be 
reduced to the form 
and the functions Wj, ... Up will possess the required property. Changing, then, each 
Uj into Xj, the expression (12.) reduces to and it only remains to express this 
in terms of zq, which, as ^^^+ 1 = 0 , is done by merely changing Xi..,Xp into 
. . . Up. 
Thus the system (11.) is reduced to 
dP 
dXpj^2 
dp 
f(H.,)^^...+(H,,) 
dV 
dx. 
•p+m 
(H. 
Mm 
dx^ *** i‘’^'dxp 
where the brackets denote that in the enclosed portion Xp+i is to be made 0, and 
Xi, X 2 , ... Xp converted into Wj, U 2 , ... 
Now this form is identical, the above conversion of letters excepted, with that of the 
system (4.), omitting that equation of the latter system by the integration of which the 
forms of ...iCp are determined. 
It follows from the above that, obtaining the integrals of AiP = 0 in such a form that 
the arbitrary constants shall represent the arbitrary values of x^ ...Xp when Xp+^=Q, and 
representing the functions which are equal to those arbitrary constants by x\, x' 2 , ... Xp, 
then if in the remaining equations A 2 P =0 ... A„jP=0 we change x^, ^ 
x-^^ ...Xp.) • • • ■^, and Xp^^ and ^ — to 0, the common integrals of the transformed 
system of^— 1 will be the same as those of the previous system of partial ditferential 
equations. In the same way a third system of ^—2 partial diiferential equations may 
be formed, and so on, till we obtain a single final partial ditferential equation which will 
have the common integrals of the parent system. By this method, which is due to 
Jacobi and Ntjtani, all the labour of the successive transformations is avoided. The 
successive integrals thus introduced are termed ‘ IIawpt4ntegraleJ 
Instead of applying the foregoing methods, general or particular, to the final system 
of partial differential equations, we may apply it to the solution of the corresponding 
final system of ordinary difierential equations. In this case they would really represent 
the method of solution known as the variation of parameters, and the conditions 
(AjAy— AyAi)P=:0 would secm'e the sufficiency of that method. If in the system of 
ordinary difierential equations (6.) we regard Xp+ 2 -> - -‘OCp+rr, as consent, we get 
= 0 , 
dXp — 'Kp)dXp+.^=z 0 . 
3 p 2 
( 13 .) 
