734 
PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
that the number of really primary forms — of forms the knowledge of which, in combi- 
nation with such known theorems, would enable us to solve all equations of the above 
type that are finitely solvable — is extremely small. It will, indeed, be a most remark- 
able conclusion, should it ultimately prove that the forms in question stand in absolute 
and exclusive connexion with the class of algebraic equations here considered. 
The following paper is a contribution to the general theory under the aspect last 
mentioned. In endeavouring to solve Mr. Harley’s equation by definite integrals, I 
was led to perceive its relation to a more general equation, and to make this the subject 
of investigation. The results will be presented in the following order : — 
First, I shall show that if u stand for the mth power of any root of the algebraic 
equation 
y n —xy n ~ l — 1 = 0, 
then u, considered as a function of x, will satisfy the differential equation 
[D> + [^D + f- l]-(~~l)^=0. 
in which e e =x , D= and the notation 
[a] 6 =a(a-l)(a-2)..(a-b+l) 
is adopted. 
Secondly, I shall show that for particular values of m, the above equation admits of 
an immediate first integral, constituting a differential equation of the n— 1th order, and 
that the results obtained by Mr. Harley are particular cases of this depressed equation, 
their difference of form arising from difference of determination of the arbitrary con- 
stant. 
Thirdly, I shall solve the general differential equation by definite integrals. 
Fourthly, I shall determine the arbitrary constants of the solution so as to express 
the mth power of that real root of the proposed algebraic equation which reduces to 1 
when x=0. 
The differential equation which forms the chief subject of these investigations certainly 
occupies an important place, if not one of exclusive importance, in the theory of that 
large class of differential equations of which the type is expressed in (3). At present, I 
am not aware of the existence of any differential equations of that particular type which 
admit of finite solution at all, otherwise than by an ultimate reduction to the form in 
question, or by a resolution into linear equations of the first order. It constitutes, in 
fact, a generalization of the form 
, a(D-2) 2 ±rc 2 
W + ~D(D-1) e w=0 
given in my memoir “ On a General Method in Analysis ” (Philosophical Transactions 
for 1844, Part II). 
