34 ; REPORT—1857. | 
In theorem 4 the author’s symbol (p,q) has a signification such as 
Poisson’s (a, 6), and if we write as before 
O(a, b) 
Opa 
(a, B) _da db db da 
O(p,9) dpdq dp dq 
(this refers of course to hypothesis 2), the theorem is, that the following 
equations subsist identically, viz., 6,@ being corresponding constants out of 
the two series 6,... and a,..., then 
(6, a)=—(a,b6)=1, 
but that for any other pairs b,a, or for any pairs whatever 0,6 or a,a, the 
corresponding symbol =0: in fact, that the constants b,... and a,... form 
a canonical system of elements. 
Theorem 5 is a theorem including theorem 4, and relating to any two 
functions u,v either of the two 2z constants or else of the 2m variables, 
(a,b)= 
where 
and which may besides contain ¢ explicitly ; it establishes, in fact, a relation — 
between Poisson’s coefficient (w,v) and the corresponding coefficient of 
Lagrange. 
Theorem 6 is as follows: viz., if g,... p,... are any 2% variables con- 
cerning which no supposition is made, except that they are connected by 
the 2 equations 
Di D(Qs = 00 Passo) 
which equations are only subject to the condition of being sufficient for the 
determination of p,... in terms of g,...anda@,..., and they may contain 
explicitly any other quantities, for example, a variable 4 ‘Then, in order 
that the 32(z—1) equations 
dp; _ ap; 
dq; dy; 
may subsist identically, it is only necessary that each of the 3n(z—1) equa- 
tions (b;, 6;)=O may be satisfied identically. 
_ Theorem 7 is, in fact, the theorem previously established in its general 
form in Liouville’s note of the 29th of June, 1853, viz., if, of the system of 
2n differential equations 
ay _ dH dp__di 
dt dp’ dt dq’ 
there be given x integrals involving the » arbitrary constants 6,..., so that 
each of these constants can be expressed as a function of the variables — 
9,++p,--. (with or without 4); then, if the in(n—1) conditions (6, 6;),=O 
subsist identically, the remaining n integrals can be found as follows :—By — 
means of the n integrals, let the 2 variables p,... be expressed in terms of 
x,...6,... and ¢, and let H stand for what H, as originally given, becomes 
when g,... are thus expressed. Then the values of p,... and —H are the — 
differential coefficients of one and the same function of p,.... and ¢; call — 
this function V, then, since its differential coefficients are all given (by the 
fete dV : ‘ ata 
iclgubig?: FRE: ‘+ 77 =—H), V may be found by integration; and itis 
therefore to be considered as a given function of p,++-and é and of the 
constants 6,.... The remaining x integrals are given by the equations 
