Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

56 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
of the second order having t=O for a singularity: it appears that the integrals are
normal in the vicinity of t=O. Their full expression is
t, f + abt2 - (abt2)2
2 A ( + p) 2.4(1 + p) (3 + p)
+ a Btb+l i1 + abt2 (abt2)2
-P ( 2(3-p) + 4(3 -p)(5 p)+- p
t- At~II{ 1]+ abt2 (abt)2 )
1 2(3 + p) 2.4(3p)(5 + p )(5+P)- }
tr = + —7 +abt { (abt2)2 ~
+ Bt 1 + 2 (1 - p) 2.4(1 -p) (3 - p)
where p = X-: in order that the solution may be satisfactory, it is manifest that p
may not be an integer, positive or negative. For the present purpose, the general
integrals must be chosen so that they vanish with t; and consequently the most
important terms in the immediate vicinity of t= O are
t, = At' +a Bt+l
i-p [^
t2 =1 At^+ +Btij
the quantities A and B being arbitrary.
If the real part of X and the real part of tu be both positive, then, when the
variable t approaches its origin not making an infinite number of circuits round that
origin, t, and t2 ultimately vanish when t= O, that is, as X and /L are not integers,
there is a double infinitude of non-regular integrals vanishing with t.
If the real part of X be positive and the real part of /u be negative, then, when
t tends to zero as before, t2 can tend to zero only if B be zero: and if B =, then
t, and to ultimately vanish when t= O, that is, there is a single infinitude of nonregular integrals vanishing with t.
Similarly, if the real part of X be negative and the real part of IL be positive,
there is a single infinitude of non-regular integrals vanishing with t.
If both the real part of X and the real part of /u be negative, then t, and t2
vanish with t only if A = O, B= 0: that is, non-regular integrals vanishing with t do
not then exist. This last result is in accordance with Goursat's result already quoted
in the introductory remarks.
It will be noticed that the parts depending upon t\ alone, when they exist, are of
the form
tl = tApl, t2 = tp2