364 Prof. G. G. Stokes, On the Arbitrary Constants [June 3, 
laborious manner, by calculating numerically from the ascending 
and descending series for the same value of the variable, and 
equating the results. 
My attention has recently been recalled to the subject, and I 
have been led to perceive that the constants in the first two forms 
of the integral may readily be connected without going behind 
the series themselves, so that the expression of the integral of the 
differential equation by means of definite integrals may be dis- 
pensed with altogether; and even if we failed to obtain a solution 
in this form the two sets of arbitrary constants could be connected 
exactly by means of known transcendents, and not merely approxi- 
mately by numerical calculation. 
The ascending, and always convergent, series treated of in the 
three papers already referred to were particular cases of one 
which, on dividing the whole by a certain power of the variable, 
has for general or (m + 1)th term 
CT (m+a)0 (m+06).. 
Qn = Tin eT Gre ee OP ceesetts fee (A), 
there being at least one more I'-function in the denominator than 
in the numerator, so that the series is always convergent. The 
connexion of the constants in the ascending and descending series 
was made to depend on two things; one, the determination of the 
critical amplitudes of the imaginary variable x, or p(cos @ +7 sin 8), 
in crossing which the arbitrary constant multiplying one of the 
divergent series was liable to change, and the mode of that 
change; the other, the determination, for some one value of @ 
lying within those limits, of that function of p to which the 
whole expression by ascending series was ultimately equal when 
p became infinite. The value of @ always chosen was such as 
to make all the terms in one of the ascending series regularly 
positive; accordingly in the series whose general term is written 
above it would be = 0, giving «=p. Now when p is very large 
the series diverges for a great number of terms, but at last we 
arrive at the greatest term, Un,» Suppose, after which the series 
begins to converge. For a ereat number of terms in the neigh- 
pourhood of Un,» “the ratio of consecutive terms is very nearly a 
ratio of equality, but the product of those ratios presently begins 
to tell. Let a and @ be two positive quantities as small as we 
please; then the number of integers lying between (1 —a)m, 
and (1+ 8) m, will increase indefinitely as p and consequently m, 
increases indefinitely, and moreover the ratio of Sw,, taken for 
values of m lying between the limits (1 — a) m, and (1+ 8)m, will 
ultimately bear to the whole series from 0 to 2 aratio of equality. 
