Yo ee Nee 
LK PERSE ee ees 
ON THEORETICAL DYNAMICS. 95 
reduced to a single differential equation between two variables, then the 
multiplier (in the ordinary sense of the word) of this last equation is MV; 
where M is the multiplier of the system, and V is a given derivative of the 
known integrals, so that the multiplier of the system being known, the in- 
tegration of the last differential equation is reduced to a mere quadrature. 
To explain the theorem more particularly, suppose that the system of given 
integrals, that is, all the integrals (except one) of the system are represented 
by p=a, g=8,..., and let w, v be any two functions whatever of the variables, 
so that p, g,.+. u,v are in number equal io the system 2, y, 2, w, ... then if 
x& eta Hae Ma +..=U 
du dv 
ve Bt YG Da et .-=V, 
the last differential equation takes the form 
Udv—Vdu=0, 
where it is assumed that U and V are, by the assistance of the given in- 
tegrals, expressed as functions of wu, v and the constants of integration. The 
multiplier of the last-mentioned equation is MV, where M is the multiplier 
of the system, and V may be expressed in either of the two forms 
J, O( 25 Ys,2, Wy.000's) 
Japa OME Lavi miseitia 2) 
és a 
V CO(asYs 2) Wye «e') 
and 
where the symbols on the right-hand sides represent functional determinants ; 
in the first form it is assumed that x,y, z, w,.- are expressed as functions of 
a, 3,.+.%,v, and in the second form that p,g,...u,v are expressed as func- 
tions of 2,y,2,w,..., but that ultimately p,g,... are replaced by their 
values in terms of the constants and w, v; the first of the two forms, from its 
not involving this transformation backwards, appears the more convenient. 
47. I have thought it worth while to quote the theorem in its general 
form, but we may take for uw, » any two of the original variables, and if, to 
fix the ideas, it is assumed that there are in all the four variables a, y, z, w, 
then the theorem will be stated more simply as follows :—given the system 
of differential equations 
dx: dy:dz:dw=X:Y:Z:W, 
and suppose that two of the integrals are p=a, g={, the last equation to be 
integrated will be 
Wdz—Zdw=0, 
where, by the assistance of the given integrals, W, Z are expressed as func- 
tions of z,w. And the multiplier of this equation is MV, where M is the 
multiplier of the system, and V, attending only to the first of the two forms, 
is given by the equation 
which supposes that x, y are expressed as functions of a, 6, 2, w. 
