2 
possibility of expressing the Boolian integrals by indefinite 
integrals. Availing ourselves of a word suggested by Mr. 
De Morgan, let us call a rational and entire function of v. 
wherein the coefficients may be any functions whatever of 
another variable x, a “ quotic.” Also let us call a rational 
fraction, whereof the denominator is the m-tli power of an 
irreducible quotic of the the ?i-th degree, and the numerator 
a quotic of a degree not exceeding mn — 1, a proper fraction 
of the ?i-th class. Then the integral, with respect to v, of a 
proper fraction of the u-tli class in general satisfies a linear 
differential equation of the (n — i)th order, wherein the in- 
dependent variable is x. But there is an advantageous 
modification of this theorem. Let 6 be a quotic of the %-th 
degree, and 3 a quotic of the (2 n — l)th degree, and, con- 
forming to the notation of my last preceding paper ( Supra 
vol. IX., pp. 8G, 87), intituled in the same way as the present 
supplement to it, let us put 
g, ( 10 ) 
Also, in (5) of the preceding paper, let us take the summa- 
tion on the sinister from a=0 to a=2n — 2 and that on the 
dexter from 6 = 0 to b=2n (2n — 2) — 1 and, further, let us 
put 
f - 
Jb (i + ey n - 2 
(li). 
Moreover let us add a term h, defined by the relation 
h = ~ { H x log (1 + 6 2 ) + Ho tan -1 0 j . . (12), 
wherein the symbols H, like F and G in (5) and (6), are 
functions of x only. Then (5) will, after these substitutions 
are made, be the convertent equation of (10), but it will be 
observed that in the present paper h is so constructed as to 
be integrable by means of logarithmic and trigonometrical 
functions, while in the preceding paper it was supposed to 
be unintegrable, save by series. Thus 2n — 2 will be the 
