FROM THE DIFFERENTIAL EQUATION. 
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a complete primitive. The corresponding general primitive is 
(x — a 2 ) + [y — -yjr (a)} 2 + z 2 = 1 
x — a + [y — 'yjr (a)} {a) = 0 
To deduce the singular solution from the differential equa 
tion (a) we have 
whence p = 0, q = 0; substituting which in (a) we find 
z = + 1. 
The above example illustrates the importance of obtaining, 
if possible, a choice of forms of the complete primitives. The 
second, of those above obtained, leads to the more interpret 
able results. It represents a sphere whose radius is unity and 
whose centre is in the plane x, y, while the derived general 
primitive represents the tubular surface generated by that 
sphere moving but not ceasing to obey the same conditions. 
The singular solution represents the two planes between which 
the motion would be confined. All these surfaces evidently 
satisfy the conditions of the problem. 
Ex. 2. Required to determine a system of surfaces such 
that the area of any portion shall be in a constant ratio 
(in : 1) to the area of its projection on the plane xy. 
The differential equation is evidently 
1 +/ + 2 2 =m 2 , 
and it will readily be found that it has only one complete 
primitive, viz. 
z = ax + V (m 2 — a 2 — 1) y + 5. 
Thus the general primitive is 
z = ax-\- V (m 2 — a? — 1) y + $ (a) 
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