561 
1907-8.] Algebra after Hamilton, or Multenions, § 10. 
The conjugate (do- /dp)' of do- /dp may conveniently be written d'o-jd'p or 
Change of independent variable in differentiations is straightforward 
in the present notation. Thus 
S dpyj = S do-yi = S >~dp.^' 
<1 P 
ci 7 d’<r , 
= Sdp v 
dp 
°r V = ^V=ViSvVi (10> 
dp 
This is analogous to 
D x = D x y.J) y , 
and of course is all that is needed in some cases of change of variable. But 
the more complicated explicit analogue of 
V y =(!>&) X l> x 
is more frequently wanted. 
By (29) § 7 this analogue is 
V' = [d a-/ dp] _1 Ko- r fri l) . SnAfei 1 * v ) 
. (11> 
In accord with our definition of dp/ do- and with (10) we may appro- 
priately assert that 
d<r = pV 
d p <rV 
( 12 ) 
The necessary and sufficient condition that the n fictors (pi v (pi 2 , .... <pi n 
are not independent, is that [0] = O. Thus the necessary and sufficient 
condition that n scalars dx v dx 2 . . . .dx n can be found such that 
— (i 1 dx 1 + i 0 dx 2 + .... + L n dx n ) = 0, 
dp 
that is, such that 
^\dy i + •... + indyn ~ 0, 
is that the Jacobian [do- /dp] vanishes. This condition then (that [do- /dp] = 0) 
is the necessary and sufficient condition that the cc’s may be varied while 
the y’s are not. If an identity holds between the y’ s it is obvious that 
such variation of the x’s is possible; and if no such identity holds, the 
x’s can be expressed in terms of the y’s, so that when dy x =....= dy n = 0, so 
also are dx x =....= dx n = 0, and such variation of the x’s is impossible. 
This establishes the well-known theorem that if [do-/dp] = 0 then 
f(Vv Vv • • • •) = and conversely. 
VOL. XXVIII. 
36 
