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Hamilton W.R. The collected mathematical papers. vol.2.. Dynamics (CUP, 1940)(600dpi)(T)(674s).djvu |
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following:
\A/C% • CvQ * • • • UU/yk ) *
(T4)
13, 14] XII...
On the other hand, in this particular and simple case, when the proposed partial differential
equation (X1) is linear, we know from the researches of Lagrange that a particular and com-
comparatively simple method may be applied, in which the equations (V2) are still useful as auxi-
auxiliary relations; and which consists in integrating those relations (V2) as an auxiliary system of
n total differential equations between the n+1 variables x, xx, .....
CALCULUS OF PRINCIPAL RELATIONS [1, 2
taking the partial differential coefficients (of the first order) of that one principal function with
respect to the initial variables al9 ..., ai and then equating these coefficients to — bl9 ..., —bi
respectively; which is my chief result respecting the properties of this principal integral S, con-
considered as depending on its limits, and my chief reason for calling that integral & principal func-
function; and for giving to that new branch of Algebra, which proposes by new methods to find and
to use the form of this principal function, the name of the Calculus of Principal Relations...
Thirdly—and this condition includes the two former ones—at the origin of the progression
or first limit of the integration (t = t±) the principal function or integral S must bear the (nascent)
ratio of unity or equality to the function formed from dS by changing the differentials dx±, ...,
dxi to the increments x1 — a1, ..., xi — ai and by changing x±, ..., xi themselves to al9 ..., at\
that is,
lim- = lim*(a1, a2, ...ai9 X±Z^1 ? ##>j ^iZ3\ ? B0)
or, in other symbols, s
lhB1)
[Solution of the partial differential equation by successive approximation.*]
[3.] We may in general consider S as a function of a±, a2,..., at-_x, ^, — , ..., ~^=^ —
xi — ai, and for small or moderate values of t —1± and oix-^ — a^ ..., xi — at we may in general
develope this function according to ascending integer powers of the small or moderate increment
xi — ai (setting aside singular exceptions) in a series of the form
S = A(xi-ai) + B (Xi - a,.J +...
+ -s— o^ — dxioyi, C3)
OX-t OXa
which may by G) be put in the form
0 = dyx8xx — dx1hy1 + .....
*i=fi(z)> E2)
and which satisfy the i initial conditions
f1(a) = a1,f2(a) = a2i ..., fi(a) = ai, E3)
and the i final conditions
/]_(x) = xx, /2(x) — x2, ..., fi{x) = xi...
+ 0/j/ (ui, Uj)8uj+ <&''' (u-,, a,)Sai^
O"
, ai)8ai,
G8)
For this purpose we may employ the relations which result from the homogeneous form of O,
namely,
G9)
(80)
and
Besides, if we take as the arbitrary multiplier A in the expressions
(see G2)):
(81)
— A^1? &c...
The system D9)-E2) is therefore a form for the complete integral of the
system of the original differential equation D4) and the n— 1 supplementary differential equa-
equations deduced by elimination from D5), and this integration of a system of several total differ-
differential equations, by means of one principal integral relation, is the chief use of such principal
relations and the reason for giving them that name...
a'n; so that we can in general eliminate the 2n quantities a[,..., a'n, a\,..., a"n
between the w+ 1 equations last mentioned and the n equations obtained from (81), and shall
47
hmpii
370 XIV...
The sum of the exponents of all these n differential equations between the n functions
xx, ..., xn is therefore
(mxl) + (kx i + 2) + (n-m'-k x i + 3) = n (i + 3) - m (i + 2) - k = 2n - 1;
conclusions therefore follow in general of the same kind as those deduced for the particular
cases above...
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