EQUATION AND ITS TRANSFORMATIONS. 
763 
finite solution corresponds in (1) to p = an integer and in Bessel’s equation to 
v = an integer +i|. The fact that Bessel’s function 3 v (x) is expressible in a finite 
form when v—i- and tlie finite expression itself, are well known, and the case is an 
important one in physical investigations ; but, so far as I know, tlie recommencement 
of the series after the zero terms has not been specially noticed in connexion with the 
subject of Bessel’s Functions. 
The eighth (and last) section contains a list of writings the contents of which are 
closely connected with the subject of the memoir, arranged in order of date and classed 
under the sections in which they are noticed. There is also in each case a short 
account of the portion of the paper for which it has been referred to, with the numbers 
of the articles in which the references occur. The section does not contain a list of 
all the papers referred to in the memoir; only those papers which are closely connected 
with it, and portions of which are, in most cases, to some extent reproduced in it, 
being included. The part of the list which relates to § VI. is intended to be supple¬ 
mentary to that section : it is not in any sense a bibliography of the symbolic solutions, 
but it probably contains references to all the more important papers on the subject. 
In the ‘ Philosophical Magazine’ for 1868 Cayley gave the four particular integrals 
P 2 , Q 3 , IP, S 2 (§ III.) of Biccati’s equation (4); and in the same journal for 1872 
I investigated the relations between these four particular integrals and the well-knowm 
particular integrals U 2 , V 3 . The results are the same as those given in § III., and the 
method is similar to that employed in § I. I afterwards found that the process of 
obtaining and connecting the particular integrals assumed a much more simple form 
when the differential equation was taken to be (l) than when it was (4); and it 
seemed desirable to re-write the whole investigation, taking (l) as the differential 
equation. This investigation forms §1.; it is similar in every respect to that contained 
in the ‘ Philosophical Magazine,’ but is much more complete. The corresponding 
results for the equations (3) and (4) are deduced in § III. 
The fact that, in the solution in series of a differential equation, if the series 
terminates but when continued recommences, the latter portion as well as the finite 
series satisfies the differential equation, was pointed out by Cayley in the ‘ Messenger 
of Mathematics’ for 1869. 
The formula (8) of §V. was published in the ‘British Association Beport’ for 1872, 
with a brief account of the process given in arts. 20, 21. The principal portion of two 
short papers, “On Biccati’s Equation” and “On certain Differential Equations allied 
to Biccati’s,” which were published in the ‘Quarterly Journal of Mathematics’for 
1871 and 1872, are incorporated in §VI. 
The memoir thus includes the results contained in several scattered notes and 
papers. In these the differential equation considered was generally Biccati’s in the 
form (4), but the advantage of adopting (1) as the standard form in preference to (4) 
is considerable. As far as the differential equation is concerned (4), which consists of 
5 F 2 
