[655
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 129
term in dt — dr
expressions x—p,y — q,z — r is an integral, or again the product of any expression of the
first set into any expression of the second set is an integral: we may take as integrals
mental theorem.
a = x 2 —p 2 , ¡3 = — q 2 , ry = z 2 —r 2 , 8 = ^ ^ , e = Z V .
r * * 1 x + p’ x+p
We have then
dt = _ a ) > that 1S > 1 “ T = lo S \ x + VO 2 - <*)} - log (x + p),
giving x + p — e tr ~ T , and thence the other quantities x — p, y + q, &c. For greater
symmetry, I introduce a new set of constants a, b, c, a', 6', c, and I write also e t ~ T =T,
rtial differential
order that the
e~ t+T = T' (where TT'= 1). We then have
x = aT 4- a'T', p — aT — a'T',
rdz), as before,
y = bT+ b'T', q =bT — b'T',
d , e
iase if r and -
6 c
is being so, we
f- H, giving the
z = cT + c'T', r = cT — c’T';
also, comparing with the values obtained as above,
a — 2 > 5 = , c=|-e,
-y)K
»'=!«, y-if. c' = j2.
We have, moreover,
ifferent process.
H = — 2 (ua' + 66' + cc') = — £ (a + (3 + 7).
76. We find
j» 2 + gfl + r 2 = # + ( ft 2 + ¿2 + <>2) p-2 + ( a '2 + ¿'2 + C '2) ^
and thence
F = A 4-J (p 2 + q 2 + r 2 ) cfa
= X + if (i - r) + 1 (a 2 + 6 2 + c 2 ) T 2 -1 (a' 2 + 6' 2 + c' 2 ) T' 2 .
We may from this obtain the expression for
dV — p dx — q dy — r dz,
when everything is variable. The terms in (dt — dr), as is obvious, disappear; omitting
these from the beginning, we have
ited upon by a
d V = d\ + (t — r) dH + (ada+bdb + c dc) T 2 — (a'da' + b'db' + c'dc') T' 2 :
also
pdx = (aT — a'T')(Tda + T'da'),
= da (aT 2 — a') + da' (— a'T’ 2 + a):
thence forming the analogous expressions for qdy and rdz, we have
)us forms, viz.
ny two of the
p dx + qdy+ r dz = (ada + b db + c dc) T 2 — (a'da' + b'db' + c'dc') T' 2
— (a'da + b'db + c'dc) 4- (a da' + bdb' + c dc),
c. x. 17