20 REPORT—1857. 
_that the solution of the partial differential equation depends on the integra- 
tion of a single system of partial differential equations. The remainder of 
the memoir is devoted to the discussion of this theory and of the integration 
of the Pfaffian system of ordinary differential equations, a system which is 
also treated of in Jacobi’s memoir of 1844, ‘Theoria Novi Multiplica- 
toris, &c. I take the opportunity of referring here to a short note by 
Brioschi, ‘ Intorno ad una Proprieta delle Equazione alle Derivate Parziale 
del Primo Ordine,’ Tortolini, t. vi. pp. 426-429 (1855), where the theory 
of the integration of a partial differential equation of the first order is pre- 
sented under a singularly elegant form. 
38. Jacobi’s note of 1837, ‘On the Integration of the Differential Equa- 
tions of Dynamics.—Jacobi remarks that it is possible to derive from 
Lagrange’s form of the equations of motion an important profit for the 
integration of these equations, and he refers to his communication of the 
29th of November 1839 to the Academy of Berlin, and to his former note 
to the Academy of Paris. He proceeds to say, that whenever the condition 
of vis viva holds good, he had found that it was possible in the integration 
of the equations of motion to follow a course such that each of the given 
integrals successively lowers by two unities the order of the system; and 
that the like theorem holds good when the condition of vis viva is not satis- 
fied, that is, when the force function involves the time (this seems to be a 
restatement, in a more general form, of the theorems referred to in the note 
of the 29th of November 1836 to the Academy of Berlin); and he men- 
tions that he had been, by his researches on the theory of numbers, led 
away from composing an extended memoir on the subject. The note then 
passes on to other subjects, and it concludes with two theorems, which are 
given without demonstration as extracts from the intended work he had 
before spoken of. ‘These theorems are in effect as follows :— 
I. Let 
Pedy ety! ay ee aU 
mn =; TI 
De ae a aye ge ae 
be the 3x differential equations of the motion of a free system, and 
4im(a? + y? 4-2" )dt=U+h, 
the equation of vis viva. 
Let V be a complete solution of the partial differential equation 
1f /dV\’, (dV\’ , (dV 
ASO feats perk saath La 
at i=) H(Z +(=) } sash 
that is, a solution containing, besides the constant attached to V by mere 
addition, 3n—1 constants a(a,,a,...a;,_,), then first the integral equa- 
tions are 
qe eNO 2 
agen. legggerend 3 
where (/3,,8,+++(33,—,) and r are new arbitrary constants: this is in fact 
the theorem already quoted from Jacobi’s memoir of 1837, and it is in the 
present place referred to as an easy generalisation of Sir W. R. Hamilton’s 
thing essential which was known in reference to the integration of partial differential equa- 
tions of the first order; he refers also to his own memoir ‘ Ueber die Pfaffsche Methode,’ 
&c., Crelle, t. ii. pp. 347-358 (1827), as presenting the method in a more symmetrical and 
compendious form, but without adding to it anything essentially new. 
