24 REPORT—1857. 
conditions, and let the forces acting on the several points in the direction of 
the axes be functions of the coordinates alone: if the determination of the 
orbits of the several points is reduced to the integration of a single differ- 
ential equation of the first order between two variables, for this equation 
there may be found, by a general rule, a multiplier which will render it 
integrable by quadratures only.” 
And for the particular case the theorem is thus stated :— 
“Given, three differential equations of the first order between the four 
quantities 9,9 Pi» Ps» 
pti ae. dT dT 
Ways Ala Ph a 0 dp,” Frito . i, 
in which Q,,Q, are functions of g,,¢_ ouly; and suppose that there are 
known two integrals a,, and that by the aid of these p,,p,, = = are 
Pi OP 
expressed by means of the quantities ¢g,,g, and the arbitrary constants 
a, 3; there then remains to be integrated an equation of the first order, 
“= 4— dg, =0 between the quantities g,,q¢,, by which is determined 
'P 
the orbit of the point on the given surface: I say that the left-hand side of 
the equation multiplied by the factor 
Pr, dpa Pa Wr 
da dB da dp’ 
will be a complete differential, or will be integrable by quadratures alone,” 
and the demonstration of the theorem is given. The remainder of the 
memoir, sections 4 to 7, is occupied by a very interesting discussion of 
various important special problems. 
45. There is an important memoir by Jacobi, which, as it relates to a : 
special problem, I will merely refer to, viz. the memoir ‘Sur I Elimination 
des Neeuds dans le probléme des trois Corps,’ Crelle, t. xxvii. pp. 115-131 
(1843). The solution is made to depend upon six differential equations, all 
of them of the first order except one, which is of the second order, and 
upon a quadrature. 
46. Jacobi’s memoir of 1844, ‘ Theoria Novi Multiplicatoris,’ &¢.—This 
is an elaborate memoir establishing the definition and developing the pro- 
perties of the “ multiplier” of a system of ordinary differential equations, or 
of a linear partial differential equation of the first order, with aopheattens to 
various systems of differential equations, and in particular to the differential 
equations of dynamics. The definition of the multiplier is as follows, viz. 
the multiplier of the system of differential equations 
ada: dy:dz:dw...=X:Y:Z:W... 
or of the linear partial differential equation of the first order 
f yo 74 wF 
pda Aa fe cw = 
een gargs een eg Capea 
is a function M, such that 
dMX ,dMY dMZ dMW 
dx dy a ga a espa 
One of the properties of the multiplier is that contained in the theorem of 
the ultimate multiplier, viz. that when all the integrals (except one) of the 
system of partial differential equations are known, and the system is thereby 
— 
