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.

76 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
and these values must, when substituted, satisfy the differential equations identically. Now
t d =, dO -
dt dt
so that
d
t (rtt) = (n + l1 + me) lmtn - m_ l~+lm-ltn.
Hence equating coefficients of glmtn on the two sides of both equations after substitution of the assumed values of t, and t2, we have
{n + (l + m - 1) I} almn - mal_, m+, n = almn
{n + (I + m - 1) } bl,, - mb_-1, m+i, n = amn + /'lmnJ
where a'ilmn, / lmn, being the coefficients of Ïq1ïr/t" in fib and 02 respectively, are linear
functions of the constants in 0b and k2, and are integral functions of the coefficients
am'n',, bl,lmf,,, such that l' < 1, mr' n m, n', l' + m' + n' < 1 + m. + n.
Assuming that the real part of e is positive but that e is not a positive integer,
we see that no one of the quantities n + (I+m-1).I can vanish if l+m+n >2.
If l = = n = 0, then a'mn = 0, /'l =0; hence
aoo0 = 0,
bo000 = Oaooo = O.
For values such that +r+n= 1, we have
O. aOl0 = 0, that is, ao00 = K,
a001 = 0,
O. aloo = O, that is, ao10 = L;
O. bolo = 0. aolo = OK,
boo001 =. a001 =,
O. bloo = 0. ao10 = OL.
In order therefore to obtain finite values for the coefficients a and b, we must have
K=O, L=O,
and then boo0, b10, are arbitrary; that is, we have
aoi0 = O, aool = O, a1oo = 0;
bo010 B, bool = O, bloo = C.
To obtain the terms of dimension two in ', rV, t in t, and t2, we require the
explicit expressions of S6 and b2: let them be
qb = attl + btt2 + Ct12 + ett2 + kt2 +...,
<2 = att1 + /tt2 + 3/t12 + etlt2 + tct22 +....