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XXXVI. On the Comparison of Transcendents, with certain applications to the Titeory 
of Definite Integrals. By George Boole, Esg., Professor of Mathematics in the 
Queens University. Communicated by Professor W, F. Donkin, F.R.S. 
Received March 16, — Read May 7, 1857. 
1. The follo^ving objects are contemplated in this paper: — 
1st. The demonstration of a fundamental theorem for the summation of integrals 
whose hmits are determined by the roots of an algebraic equation. 
2ndly. The application of that theorem to the problem of the comparison of algebraic 
transcendents. 
The immediate object of this application will in each case be the finite expression of 
the value of the sum of a series of integrals, lfX.dx, the differential coefficient, X, being 
an algebraic function, and the values of x at the limits being determined by the roots of 
an algebraic equation. 
3rdly. The application of the same theorem in a new, and, as is conceived, more 
remarkable line of investigation, to the comparison of functional transcendents. 
The terms ‘algebraic’ and ‘functional’ are not here used by way of logical division 
to indicate classes of transcendents wholly distinct, but the term functional transcendent 
is simply employed to designate an integral ^ILdx in which X involves an arbitrary 
symbol of functionality. 
Under this thhd head of the comparison of functional transcendents will fall the most 
important special result of the entire investigation. A case will arise in which, without 
any hmitation of the functional symbol, the several integrals included under the form 
lfK.dx will close up, if the expression may be allowed, into a single integral taken 
between the limits — oo and co. The result is a very remarkable theorem of definite 
integration, fruitful in important consequences. In its general form, this theorem is, I 
believe, entirely new. A particular case of it was discovered by me several years ago, 
and was published without demonstration in the Cambridge and Dublin Mathematical 
.loumal*, and in Liouville’s Journal de Mathematiquesf. A memoir by Cauchy on 
integrals taken between the limits 0 and ooj, contains also a very limited case of the 
same theorem. It appears there, however, as an isolated result, quite apart from the 
doctrine of the comparison of transcendents. 
In the concluding sections of this paper I shall apply the results of this part of the 
investigation to the extension of the theory of multiple definite integrals. 
As respects the methods and processes which will be employed in this paper, the only 
* Vol. iv. p. 14. t Tom. xiii. X Bxercices, vol. i. p. 54. 
MDCCCLVII. 5 E 
