DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
329 
is canonical. Let us assume then that a complete set of normal integrals 
hi, ... hn of this latter system is known, so that we have 
&c., yi, &c., t), &c., yi, &c., t). 
The assumption of these last equations to represent the solution of the complete 
system (90.) is simply a transformation of variables, {a^, &c., h^, &c. being the new 
variables) ; it is also a normal transformation, since the equations connecting the 
new and old variables may be put in the form (see Theorem VII. art. 49, and art. 62.) 
^_7 
dxi dtti 
where X (the modulus of transformation) is a function of ... a^, ... t. The 
dX. 
function T" of art. 62 is now obtained by expressing — ^ terms of a^, ... hi, ... ; 
dX^ 
but since Z is — expressed in terms of Xj, &c., y^, &c., it follows that when Z is 
expressed in terms of the new variables «!, &c., b^, &c., it becomes identical with 
Now if the process of art. 63 be followed, mutatis mutandis, it will be seen that in 
the present case we obtain 
, d'^ dTi /dO, di/j\ 
dbi dbi dbij’ 
in which expression the first two terms destroy one another, and the remaining term 
is evidently the differential coefficient of Q with respect to bi, taken so far as Q con- 
tains bi independently of those functions which were exempt from differentiation in 
forming the original differential equations (90.). Similar reasoning applies to the 
expression for 6-. 
As this result will be useful, I shall enunciate it separately as 
Theorem XL— If the original system of differential equations be formed by treating 
certain functions,^, q, r, ..., contained in the disturbing function Q, as exempt from 
differentiation with respect to x^, &c., yi, &c., the equations which determine the 
variations of any set of normal elements Ui, &c., Z»i, &c. 
' 
dbi ‘ dai 
on condition that jo, q, r, ... be treated, in forming these equations, as exempt from 
differentiation with respect to a^, &c., b^, &c. 
[It is important to recollect, that after these equations are formed, p, q, r, &c. are 
to be expressed in terms of a,, &c., b^, &c., and in the integration of the system 
&c., bi, &c. are to be treated indiscriminately as variables, whether they originally 
entered through p, q,r, ... or not]. 
The Theorem XL may also be immediately obtained from the general equations 
(E.) of art. 52 (in which it is to be remembered that Z includes the disturbing 
