, faz 
ON THEORETICAL DYNAMICS. 11 
dV ,aV_ , aV_ , 
im m iy Y > nt MZ, 
dV , dV —m , d Nisisuspge 
da db dd 1 
dV _ 
2) iio 
and, considering V as a known function of 2, y, z, a, b,c, H, the elimination 
of H gives a set of equations which are in fact the integral equations of 
the problem, viz., the first three equations and the last equation give equa- 
tions containing 2, y, z, 2',y', 2',é and a,b, c, that is, the intermediate inte- 
grals ; the second three equations and the last equation, give equations 
containing 2, y, 2, t, a,b, ¢, a', b', c', that is, the final integrals. 
The function V satisfies the two partial differential equations 
1 ffdV\?, (dV? . (dV?) _ 
ao (hee Ca) kaa) pe 
1 dV\*  /dV\? , /dV\? 
fs a fees ag el = H; 
mat (a +(%) +(3) I Von 
which, if they could be integrated, would give V as a function of 2,y, 2, 
a, b,c, H, and thus determine the motion of the system. 
_ 23. Hamilton’s principal function S.—This is connected with the function 
V by the equation 
V=/H+S; 
or, what is the same thing, the new principal function S is defined by the 
equation 
t 
s=( (T+U)de; 
0 
but S is considered (not like V as a function of x,y, z,a,6,c; H, but) as 
a function of 2, y, z,a,6,c,t. The expression for the variation of S is 
OS= — Hée+ m(a'dx + y'dy +-2'dz)---m(alea + b'eb +-cléc) 
which is equivalent to the system 
pa ma’ 3 _ my’ a mz’ 
ole Th eee tile 8 oe 
ds , dS , aS , 
Peat io ea by ine’, 
ds 
—=—H; 
dt i 
the first three and the second three of which give, respectively, the inter- 
mediate and the final integrals; the last equation leads only to the expres- 
_ sion of the supernumerary constant H in terms of the initial coordinates 
a, b,c, and it may be omitted from the system. 
_ The function S satisfies the partial differential equations 
dS, 1 f/dS\?. /dS\? he 
tam (ae) +(a) (3) re 
dS 1 f/dS\*, (dS\?, /d8\21__ 
+5. ta) + (a5 +(Je) f =U0s 
