DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
303 
all given ^by the equations ^=3/,, — (Z)^, X may be found by simple 
i[ue- 
gration, and is therefore to be considered a given function of x^, ... x^, a,, ... a„ and t. 
The remaining n integrals are then given by the n equations 
dX , 
... bn being n new arbitrary constants. 
[On the relation between this theorem and the theories of Sir W. R. Hamilton 
and Jacobi, see arts. 15-20.] 
50. Other results established in the former part will be referred to as occasion may 
require. To the theorems enunciated in the preceding article, the following may 
now be added. 
Returning to the equations (50.), (54.), (55.), we may observe, that if, in the Jirst 
members of (55.), x,, 3/^ be supposed expressed in terms of «i, &c., &c. and t, then 
dx' dv' • 4 
may be written instead of x', 3/- ; since on this hypothesis the total differential 
coefficients of yi are obtained by differentiating with respect to t as it appears 
explicitly. We have therefore 
dxi 6?Z diji d7i 
dt dyi dt dxi 
where the first members refer to Hyp. I., and the second to Hyp. II. But since the 
equations (50.), (54.) involve a, b, exactly in the same way as they involve x,y, it is 
obvious that the same reasoning which leads to the equations just written, would 
lead, mutatis mutandis, to the following, which may be considered as an addition to 
the system of equations (57.) (Theorem III.): 
dui dTj dbi d7i s 
dt dbi dt dui 
In these equations, bi in the Jirst members are supposed to be expressed in terms 
of the variables {Hyp. II.), whilst in the second members x^, &c., 3/1, &c. are supposed 
to be expressed in terms of the constants and t {Hyp. I.). As before, Z= — but 
in (60.) Z is differently expressed, being what the Z of (55.) becomes when x^, &c., 
3/,, &c. are expressed according to Hyp. I. 
It is to be remembered that all consequences deduced from the form of the system 
(50.), (54.) belong to the system of equations, obtained as in Theorem VII., which 
express the solution of the differential equations (I.). Such a solution will be called, 
as before, a normal solution ; and the system of equations obtained by expressing 
fl,, &c., ii, &c. in terms of the variables and t, will be called a system of normal inte- 
grals. (See art. 20, and the note to art. 29.) 
51. Let «i, &c., hi. See. be called, as before, elements. If then c be any function of 
the elements, when the latter are expressed in terms of the variables and t 
