EQUATION AND ITS TRANSFORMATIONS. 
765 
but as if they were essentially different, and the processes of solution have been 
applied to them independently. Also many of the forms have been re-discovered 
several times; and it therefore seemed to be worth while to collect together, as in 
§ YI., the different symbolic formulae, and exhibit the nature of the relations between 
them. 
Although the equation (1) is connected with Bessel’s equation by such simple 
relations, the methods of treatment of the two equations by mathematicians have been 
very different. In the case of (1) and its transformations (2), (3), (4), the purely 
analytical part of the theory and the forms of the solutions have chiefly attracted 
attention ; while in the case of Bessel’s equation the expansion of the results in series 
suitable for calculation has been one of the main objects. The theories of the two 
equations have been developed from very different points of view : the one has been 
considered in reference to the methods of solution and the peculiarities already referred 
to, and the other has been considered almost wholly in connexion with the functions 
which satisfy it, and their applications in astronomy and physics. It is curious that 
two such very distinct classes of analytical investigation should have been formed 
having reference to differential equations so closely connected. 
It is proper to remark here that in the differential equation (1) and throughout the 
memoir the constant a may be put equal to unity without loss of generality. It was' 
found to be desirable to retain it, as there is some advantage in having present in the 
solutions a letter whose sign can be changed at pleasure, and also because the transition 
to the differential equations 
clH 
t d - a~u — 
pfp + 1) 
u, 
&c., 
(i.e., in which the sign of a 3 is changed) is thus rendered somewhat more convenient. 
The ordinary differential equations (1), . . . (4) are considered throughout, and no 
reference is made to the corresponding partial differential equations 
clho a dht j)(p + 1 ) 
-rr “ «Va ~- o — u 
dx l dy z x 2 
&c., 
the solutions of which may be deduced in the usual manner by replacing a by a—, 
and c x e ax and c 2 e ax by and xft(//-{-ax). No point of interest arises in connexion 
with this transition. 
Following the notation usually adopted in connexion with the differential equa¬ 
tion (1), i is used throughout to denote a positive integer. The expression v /(—1), 
which occurs only towards the end of § VI. and in § VII., is denoted by % . 
