OF THE FIRST ORDER. 
Therefore replacing y x by y, 
Now the principle of continuity demands that in order 
that the solution of a difference-equation of the first order 
may merge into a solution of the limiting differential equa 
tion, the value which it gives to the above expression for 
tan 6 should, as Ax approaches to 0, tend to become infini 
tesimal ; since in any continuous curve or continuous portion 
of a curve tan 6 is infinitesimal. Again, that the above ex 
pression for tan 6 should become infinitesimal, it is clearly 
A 2 « 
necessary and sufficient that should become so. 
35. The application of this principle is obvious. Sup 
posing that we are in possession of any of the complete 
primitives of a difference-equation in which Ax is indeter 
minate, then if, in one of those primitives, the value of Ax 
A 2 ?/ 
being indefinitely diminished, that of tends, independ 
ently of the value of the arbitrary constant c, to become infini 
tesimal also, the complete primitive merges into a complete 
A 2 ?/ 
primitive of the limiting differential equation; but if 
tend to become infinitesimal with Ax only for a particular 
value of c, then only the particular integral corresponding to 
that value merges into a solution of the differential equation. 
86. "We have seen that when a difference-equation of the 
first order has two complete primitives standing in mutual re 
lation of direct and indirect integrals, each of them represents 
in geometry a system of envelopes to the loci represented 
by the other. Now suppose that one of these primitives 
should, according to the above process, merge into a com 
plete primitive of the limiting differential equation, while 
the other furnishes only a particular solution; then the 
latter, not being included in the complete primitive of the 
differential equation, will be a singular solution, and retain