446 NOTE ON THE THEORY OF LINEAR DIFFERENTIAL EQUATIONS. [862 
(that is, nearer + go ) than any other exponent: we have thus a highest power of t, 
the whole coefficient for which must vanish, viz. we obtain an equation of two or 
more terms giving for A a value or values not = 0; in the term At a in question, 
a is in general positive, but it may be = 0, or be negative. The leading term At a 
being found in this manner, the law for the exponents of the subsequent terms is 
frequently at once apparent; but, if necessary, we write z = At a 4- A't a ~ a ', and determine 
in like manner the exponent a — a! and the coefficient A'. Proceeding in this 
manner until the law of the exponent becomes apparent, and then by the method of 
indeterminate coefficients, we finally arrive at a series 
z = At a + A't a ~ a ' + A"t a ~ a '~ a '' + ..., 
in descending powers of t: the exponents are integral or fractional, but the number 
of positive exponents is always finite. There is either a single series or it may be 
two or more series; but the number of series is at most = to, and when it is less 
than to, then the coefficients in the several series or some of them will contain 
radicals, and by giving to each radical its different values, the system of series will 
determine to different values of z. 
7. The curve meets the line infinity (s=0) in the point K counting as k inter 
sections, and in to other points, some or all of which may coincide with K; the forms 
of the series depend on the configuration of the m points, or say on the relation of 
the curve to the line infinity. Thus suppose k = 0, and further that the to points are 
all of them distinct, that is, let the curve be a curve of the order to meeting the line 
infinity in to distinct points, or, what is the same thing, having to asymptotes, no two 
of them parallel; we have in this case to series, each of the form 
„ C D 
z — At B — + ..., 
Tj Tj 
where the several coefficients A have distinct values. 
8. In the foregoing case A is determined by an equation of the order to, having 
unequal roots; and taking for A any root at pleasure, the remaining coefficients 
B, C,... are each of them linearly determined. If the equation has two equal roots, 
this may correspond to the case of two parallel asymptotes (the curve has here a 
node at infinity); the coefficients B, C,... will in this case depend on a quadric 
radical, and by giving to this radical its two values, we obtain the two series 
z = At + B + — + ..., 
u 
z = At -f B' + ——f- ..., 
Tj 
corresponding to the two equal roots of the equation. But if instead of a node at 
infinity we have the line infinity a tangent to the curve, then instead of the two 
parallel asymptotes, we have an asymptotic parabola; the series assumes a new form, 
viz. it contains terms in P, t~*, ... ; the coefficients of the integer powers have