DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
89 
7^r 
Final!}', then, if X be a complete solution, the assumptions ~=b t involve as a 
consequence the relations ^=^5 and these again involve ?/•= — where?/; stands 
f dX 
for ~j — 
dxi 
In thus applying Jacobi’s demonstration I have slightly altered its form, in order 
to bring more prominently into view the necessity for X being a complete solution. 
16. It is obvious, from the considerations given in art. 13, that instead of the 
equation (X.) of the last article, we might employ anyone of the analogous equations 
obtained by distributing the variables as explained in the article referred to, and then 
dQ 
writing for in the expression for Z. The function Q will be a “ principal func- 
tion.” In particular, if we take the equation 
d Y AdY dY 
dt J\dyt drj’ 
Vn, t 
}= 0 , 
any complete solution will give the integrals of the differential equations (I.) by means 
of the system 
dY dY . 
■=X i9 
dyi da t 
The whole number of partial differential equations from each of which a “ principal 
function” can be obtained, will obviously be 2 ". The relations between these dif- 
ferent principal functions will be apparent from the conclusions of art. 4*. 
17. If x 1} x 2 , ... x n represent all the independent coordinates (of whatever kind) in 
any ordinary dynamical problem, and T the expression for the vis viva-f in terms of 
x 1} &c., x[, &c., the equations of motion are, as is well known, 
( d ^)'_dT_dV (T , 
\dxiJ dxi dxf ' ' 
where U is a function of x 1} ... x n , which may also contain t explicitly, but not x\, &c. 
Lagrange, to whom these formulse are due, was also the first to employ the expressions 
dxi 
7 as new variables, instead of But Sir W. Hamilton first showed that this sub- 
stitution (^putting — would reduce the n equations (T.) to the 2 n equations of 
the first order of the form (I.), art. 14. His demonstration, however^;, depends upon 
the circumstance that T is, in dynamical problems, necessarily homogeneous with 
respect to x\, .... x n , and I am not aware that any other case has hitherto been con- 
templated. 
The investigations of the preceding articles will however enable us to apply a 
* Compare Sir W. Hamilton’s expressions. Philosophical Transactions, 1835, p. 99, art. 5. 
t I here adopt, what I hope will be universally adopted, the suggestion of Coriolis and Professor 
Helmholtz, that the definition of vis viva should be half the sum of products of masses by squares of velocities. 
1 Philosophical Transactions, 1835, p. 97, art. 3. 
MDCCCLIV. N 
