1889.] = which multiply Semi-convergent Developments. 363 
but which can of course be considered quite apart from their 
applications. These functions present themselves as the com- 
plete integrals of certain linear differential equations, or it may 
be as definite integrals which lead to such ditferential equations, 
of which they form particular integrals; and as of course the 
theory of the complete integrals includes that of any particular 
integrals, the subject is best regarded from the former, and more 
general, point of view. 
The independent variable was taken as in general a mixed 
imaginary, and the complete integral was expressed in two ways, 
either by ascending series which were always convergent, or by 
exponentials multiplied by descending series which were always 
divergent (except in very special cases in which they might 
terminate), though when the divergent series were practically 
useful they were of the kind that has been called semi-convergent. 
In either form of the complete integral, the arbitrary constants 
appeared as multipliers of the infinite series (of the ascending or 
descending as the case might be), or it might be, in part, of a 
function in finite terms. The determination of the arbitrary 
constants, a thing in general so easy, formed here one of the chief 
difficulties ; and the capital problem may be stated to be, to find 
the relations (linear relations of course) between the arbitrary 
constants in the one and those in the other of these two forms of 
the complete integral. 
In the papers referred to, this was always effected by means 
of a third form of the complete integral, in which it was expressed 
by definite integrals, their coefficients forming a third set of 
arbitrary constants. The first two forms of integral were useful 
for numerical calculation, the one or the other being preferred 
according as the modulus of the independent variable was small 
or large; the second form indeed could be used only when the 
modulus was sufficiently large, so that the adoption of the first 
form in that case was not merely a matter of preference; the first 
form could theoretically be used in any case, but the numerical 
calculation would become inconveniently or even impracticably 
long if the modulus were large. The third form was not con- 
venient for numerical calculation, and was used only as a journey- 
man solution, for connecting the arbitrary constants in the first and 
second forms of integral with one another, by connecting them 
each in the first instance with the set in the third form of 
solution. JI remarked that in the event of our not being able to 
obtain a solution of the differential equation in the form of 
definite integrals, the use of the first two forms of integral would 
not therefore fall to the ground; the linear relations between the 
arbitrary constants in the first and those in the second form could 
still be obtained numerically, though in an inelegant and more 
