316 
PROFESSOR DONKIN ON THE 
of differential equations (I.) be transformed by the introduction of new variables 
li, ... jji, connected with the original variables ... x^, 3/1, by the equa- 
. L , 3/1 
tain t explicitly, then the transformed equations are 
tions -j-z=Xi, where P is any function of | 
(li/i 
, 3/„, which may also con- 
drii 
(79-) 
in which O is defined by the equation 
o=z- 
dt ’ 
(The substitution of the new 
See art. 63.). 
and is to be expressed in terms of the new variables. 
dP . 
variables in is to be made after the differentiation. 
dP 
Corollary . — If P do not contain t explicitly, -^=6 and 0=Z ; so that in this case 
the transformation is effected merely by expressing Z in terms of the new variables. 
65. It follovvs from {77 •)> art 63, that if fg be any two integrals of the system (I.), 
the value of [/*, g] is the same whether it be derived from these integrals in their 
original form, or similarly obtained from the same integrals after transformation by 
the introduction of the new variables. And consequently ifn integrals Ui, Uj, ... a„ of 
the original system be given, which satisfy the conditions [a;, aj]=0, they will 
continue, after a normal transformation, to satisfy the analogous conditions, so that the 
method of finding the remaining integrals given in Theorem VII. art. 49, will also 
continue to be applicable. We had an instance of this in the case of the problem of 
central forces (art. 27.), where the above conditions were found to subsist after the 
transformation from rectangular to polar coordinates. (It will be shown presently 
that every transformation of coordinates is a normal transformation^) 
66. It was shown in Part I. (art. 18.), that if W be any function of x^, x^, ... x^, 
x\, x^, ...x'n (which may also contain t explicitly), the system of n differential equa- 
tions of the second order 
(S)'=S <~> 
may be changed into a system of 2n equations of the first order of the form (I.), 
of it, deduced Jacobi’s form of the method of the Variation of Elements (namely, the equations (68.), art. 54), 
from the similar form of Lageange, in which the elements are the initial values of the variables. It will appear 
in the sequel that the extension to the case in which P may contain t, is of importance. If the expression were 
not already appropriated, I should have proposed definitively to call P the “ modulus of transformation;” and I 
shall use this term provisionally in the present paper, not being able to suggest a tolerable substitute. After all, 
as the word “ modulus ” itself is used without confusion in very different senses according to the subject matter, 
there is, perhaps, no reason why a similar liberty should not be allowed in the use of the proposed expression. 
