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Hamilton W.R. The collected mathematical papers. vol.2.. Dynamics (CUP, 1940)(600dpi)(T)(674s).djvu |
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Ivv \ Ft//
f (x )
y1F'(y1) + ...+ynF'(yn)= ~~uj' (u1) + .S+uJ' {uj )
we can now easily eliminate all the partial differential coefficients of the function F from the
auxiliary system of total differential equations (V2) and (W2), and may write these equations
as follows:
and
(Q4)
•••+«»/'
It results from the general theory explained in the 10th article that the equations of this system
(P4) and (Q4) must be capable of being reduced to 2n — 1 distinct differential equations between
2n variables; and, accordingly, we may consider any one of the 2n equations of this system as
being a consequence of the 2n— 1 other equations of the same system, because the equations
(A4) and (B4) give this differential relation, analogous to (I4) and deduced by a similar process:
0 =/' D>)...
For it is easy now to perceive (from the investigations of the last article) that in order to inte-
integrate any proposed partial differential equation of the 1st order
0 = F(c/>,xl,...xn,yli...yn); (X1)
HMPII 42
330 XII...
(8)
/
We shall call the integral S=$dS, determined in this way, the principal integral* of the given
element dS, or of the function O, in equation A) and shall denote it, for distinction, by the
symbolic expression
#= \dS= O(xl5 ..., xi, dx±, ..., dx{), (9)
drawing a stroke under the sign/ of integration...
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 +...
Xk
And we may in general develope this equation B4) by Taylor's theorem as follows:
0 = Y0 + Y1 + Ya + Y8 + &c, B5)
in which xFn is homogeneous of dimension n with respect to the increments x1 — a1, ..., xi — ai]
and then may deduce from it the following indefinite series of separate equations in partial
differential coefficients of the first order,
, 0 = Tl5 0 = T2, 0 = Y3, &c...
+ -s— o^ — dxioyi, C3)
OX-t OXa
which may by G) be put in the form
0 = dyx8xx — dx1hy1 + .....
D4)
To transform this expression for the first correction S2 of the first approximate value S1 of S,
we may observe that the equation C9) gives, when varied with respect to all the quantities
x±, &c...
, Ccu* -f- Z — (JU , - ...,
dz \ x — a x — a x — a x — at
f ...+d>'(ai)(xi-ai)}, E8)
O'^), ..., O'(a^) being here formed by varying O \a1,a2, ...,ai9 ———, ———, ...,— M as
\ x — a x — a x a j
if— , etc...
Without
actually performing this elimination (which we cannot perform while we leave the form of
undetermined) we may still deduce these differential coefficients as follows:
The complete variation of S± is, by F8) and G0),
8S1 = v18u1 + v28u2 + ...+vi8ui + ^f(a1)8a1 + ^f(a2)8a2 + ...+(^f(ai)8ai; G2)
and comparing this with the variation of the expression F9) we find
0 = u18v1 + u28v2+ .....
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