[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