An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

AN EQUATION WHICH BECOMES ILLUSORY.

109

and then putting y = 0, we have
1 1
C + D- + E =0.
U0 U0
Hence if E is not zero - cannot be zero; and therefore
u0 1
u0 cannot be infinite: but if E = o, then one value of- is
U0
zero, and therefore one value of u, infinite: if D also = o, both
values of- are zero, and therefore both values of u0 infinite.
U0
If C, D, and E be all zero, then in the equation (2),
dividing out x2, and putting x = o, we have
F + Guo + Huo2 + Iu03 = 0..
which equation gives us the finite values of u0. To obtain the
infinite values we have, as before, the equation
F- + G -+ H —+I=o.
U0 Uo U0
It appears from these equations that there are in this case
dy 1
3 values of when x = 0. If I= 0 one of the values ofdx U
is zero, and therefore one of the values of u0 is infinite: if
I = o, H = 0, two of the values of - are zero, and therefore
U0
two of u0 are infinite: if G = 0 also, the three values of U0
are infinite: if I o and F = o, a value of uo is 0, and a value
of - is o, and either of the equations becomes G + Huo = 0,
o GC
which gives u0 = -; hence, in this case, the three values
G
of U are 0, o, and -
H'
If F, G, H, I, are all zero, we must divide out x3 in
equation (2) and proceed as before; and thus we may in all
cases obtain an equation for determining all the values of u0.