655] 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
More symmetrically, we have the solution 
V = X — \(x 2 + y 2 + z 2 ) + 
(ax + by + cz) 2 
a 2 + b 2 + c 2 ’ 
as can be at once verified. 
131 
78. In the same particular case H= 0, introducing the corresponding values 
p 0 , ( b, n, «o, Vo, zo, we find a very simple expression for V — V 0 , as a function of 
x, y, z, x 0 , y 0 , z 0 . We have, writing T 0 = e ta ~ T , T 0 ' = e~^ +T , and therefore T 0 T 0 '=1, 
x 0 = aT 0 + a'T 0 ', p 0 = aT 0 - a'T 0 ', 
y 0 =bT 0 + b'T 0 ', q 0 = bT 0 — b'T 0 ', 
z 0 = cT 0 + c'TJ, r 0 = cT 0 - c'T 0 \ 
and thence 
x - = a (T - T 0 ) + a' (I - i), = (T - T.) (a - , 
® + m,-a(J'+r,) + o'(i + yJ. =(T-n)(« + ^r). 
Forming the analogous quantities y — y 0 , &c., we deduce 
(x - x 0 ) 2 + (y- y 0 ) 2 + (z - z 0 ) 2 = (T - T 0 ) 2 1a 2 + b 2 + c 2 + (a' 2 + b' 2 + c' 2 ) i , 
(x + x 0 ) 2 + (y + y 0 ) 2 + (z + zo) 2 = (T+ T 0 ) 2 ja 2 + b 2 +c 2 + (a' 2 + b' 2 + c 2 ) ^ j . 
But we have 
v- V, = i j(« s + v + (?) (T- - T,*) - (a'’ + V* + c' ! ) (L _ J_)j 
= i(T* - T,*) |o* + 6« + c* + (a'» + + c'*) A_l, 
and hence the required formula 
V-V 0 = \ V{0 - Xo) 2 + (y- y 0 ) 2 -t-(z- ¿o) 2 } V{0 + ^o) 2 + (y + 2/o) 2 + 0 + z 0 ) 2 }, 
or, say, for shortness, 
= 4 VCR) VOS). 
79. We ought, therefore, to have 
\ d V(-R) V($) =p dx +qdy +rdz—p 0 dx 0 — q 0 dy 0 — r 0 dz 0 , 
where p, q, r, p 0 , q 0 , r 0 denote as above, and consequently 
p 2 + q 2 + r 2 = x 2 + y 2 + z 2 , p 0 2 + q<? + r 0 2 = x 0 2 + y 0 2 + z 0 2 .