328 
PROFESSOR DONKIN ON THE 
pendently ofp, q, r, &c. ; and therefore the term represents the differential coefficient 
^ taken so far as Z contains independently of p, q, r, &c. The same reasoning 
applies to the corresponding term in the value of rii. The theorem is thus established. 
It is evident that it may be combined with that given in art. 71, where other func- 
tions analogous to p, q, r, ... are introduced by the modulus of transformation P. 
If we call the form of the system of differential equations (89.) canonical when the 
differentiations of Z with respect to x^, &c., y^, &c. are total, we might eall it pseudo- 
canonical when Z contains functions of x^, &c., y^, which are exempt from differentia- 
tion in forming the differential equations. 
In like manner, if we call a transformation of variables normal, when the differen- 
tiations of the modulus P (equations (73.), art. 62.) with respect to li, &c., 3 / 1 , &c. are 
total (as in art. 62.), we might call the transformation pseudo-normal when P contains 
functions of the variables which are exempt from differentiation' in forming the equa- 
tions of transformation (as in art. 71 -)- 
Adopting these designations, we may enunciate the following general theorem of 
transformation : — 
Theorem X. — If a pseudo- canonical system be transformed by a normal or pseudo- 
normal substitution, the transformed equations are also pseudo-canonical, and may be 
formed by the rules applying to normal transformations of canonical systems, provided 
that the functions wtiich are originally exempt from differentiation with respect to 
the variables, be continued exempt to the end of the process ; but if such functions 
occur in the modulus of transformation P, they are subject to total differentiation 
with respect to t in forming the term (See art. 71-) 
[With respect to this theorem there is one important remark to be made. If u, v 
be any two functions of x^, &c., 3 /i, &c. (with or without p, q, r, ... and t), the equation 
rdu dv 
du dv\ ^ 
rdu dv 
du dv\ 
\dyi dxi 
dxi dyi) 
~d^i diii) 
(art, 63.) 
is now only true on condition that the substitution of the actual values ofjo, q,r, ... 
in terms of the variables be not performed till after all the differentiations.] 
75 . The theory of the variation of elements affords an interesting example of the 
theorem given in the last article. Consider the following system of differential 
equations. 
, d7i d£l , 
dZ dfl 
dxi dxi 
(90.) 
where in the differentiations are total, but n is supposed to contain functions 
of Xj, &c., y^, &c., which are exempt from differentiation in forming the above equa- 
tions. The system 
dZ , dZ 
