340 
DERIVATION OF THE SINGULAR SOLUTION 
Hence q — (s -2 — 1 — p 2 )*, and the auxiliary system (5), Art. 7, 
"becomes, on substitution and division by (z~ 2 — 1 —_p 2 )*, 
dx 
p {z 2 — 1 — p 2 ) 
dz z s dp 
p 
.(b). 
From the last two members we have on integration 
c(l-z 2 )* 
p = — - . 
* z 
Substituting this, with the corresponding value of q derived 
from (a), in the equation dz =pdx + qdy we have 
c (1 — z 2 ) l dx . 2 ., (1 — z 2 )* , 
dz = -h -1 + (1 - c 2 f j- L dy, 
integrating which in the usual way, we find 
(1 — z 2 ) 1 = — cx — (1 — c 2 ) l y — c, 
or, changing the signs of c and c, 
(1 — z 2 ) 1 = cx — (1 — c 2 ) l y + c (c), 
which is a complete primitive. The corresponding form of the 
general primitive will be 
(1 — z 2 ) i = cx — (1 — c 2 )*y + (j) (c) 
0 = x — c (1 — c 2 )~ h y + (j> 
from which c must be eliminated. 
3) \ 
'(c) y 
(d), 
But another system of solutions exists; for from the first, 
third, and fourth members of (b) w r e may deduce 
pdz + zdp + dx — 0, 
whence pz + x = a, from which, and from the given equation 
determining p and q, we have to integrate 
, a — x 7 {1 — (a — x) 2 — z 2 )} 1 , 
dz = dx + - - dy. 
z z ^ 
The result is 
(x-a) 2 + (y -b) 2 + z 2 -l (e),