2 55 
the calculus of functions. 
that the constant quantity introduced by integration, is not 
perfectly arbitrary, it must be determined so as to make the 
equation between oc andy fulfil the equation ( a ). If for in- 
stance, we assume <p (~ j to be equal to a we should find 
the equation of the curve to be 
y = (a + 9c) c 
c being the constant introduced by integration, and on sub- 
stituting this value of y in ( a ) we shall find c= o, s o that 
y — (a ^ x)o 
Let us suppose a to be infinite and equal to-^, then we have 
Xj c ~b cx •==. b, since c = o 
which is the equation of a straight line parallel to the axis 
of the at’s, which in fact agrees with the conditions of the 
Problem. If we suppose j = a constant quantity 5 
we should find 
oc=z cVy t —a 2 
this value being substituted in ( a ) gives for determining c the 
equation 
c z (r 2 -f- i ) = o 
whence c=o and c- y/— l, using this latter value we have 
x~ %/ • — 1 x v/y — ct=.\/ a* — y z 
which is the equation of the circle, and it is obvious, that this 
curve satisfies the conditions. 
It is very necessary to attend to this mode of determining 
the constants, as we should otherwise meet in the solution 
with many curves which do not satisfy the conditions ; thus 
in the last example, the curve is apparently an hyperbola, 
but owing to the constant becoming imaginary, it is in fact a 
circle. 
