748 PEOFESSOE BOOLE ON THE COMPAEISON OP TEANSCENDENTS, WITK 
«!, a^, . . a,, being the independent constants in the equation (1.) by which the limits are 
determined, or the form 
Xd:r=-^(Xi, x^, . . x„)-\-C, (6.) 
■>^(Xj, X 2 , . . x„) being a function, and manifestly a s^unmetrical function, of the limits 
x„ X 2 , . . Xn- Either of these forms may be converted into the other by means of (1.). 
but the fii’st is the one to which we shall give the preference. 
We may remark, that if in the equation (1.) «i, lEe values of x. 
determined by the solution of that equation, will vary also. For each set of values of 
a,, ^25 • • there will exist a simultaneous set of values of x. We may in this way 
consider the variables x in the several integrals under the sign 2 as always, in the course 
of their transition from the lower to the upper limits of integration, determined by the 
roots of the equation 
E(.r, «i, «2? • • = 
According to this more general view, «i, - • dr become a set of variables with which x 
is connected by the above equation, but the variation of a; in each integral represents 
only the variation of one root of the equation. And the determination of the values of 
X at the upper or at the lower limits of integration by the solution of that equation, 
particular values being assigned to ^i, is only a special case of the deteimina- 
tion of the simultaneous values of the variable x. 
4. Thus the problem with which we are concerned may be more briefly expressed in 
this form. Beqiiired the value of the expression 2jX(?x, the simultaneous values ofy. in 
the several integrals being determined by an equation of the form 
E(^, «i, «2, • • = (7.) 
in which Uj, a 2 , . . a^ are variable quantities, by the assigning of particular values to which 
in the solution of the equation, the particular values of x at the limits of integration are 
determined. 
The solution of the above problem is to be effected by giving to the expression 'IfXdx 
the equivalent form j IXdx, transforming XXdx into a complete differential uith respect 
to the variables a^, a^, . . a,., and then effecting the integration. For this reason I shall 
designate (7.), or, as it may for convenience be written, E=0, as the ‘ ti'ansforming equa- 
tion,’ except when it is employed to determine the limits, in which case the designation 
of ‘ equation of the limits ’ will be preferable. 
For the solution of the problem, as thus stated, it is usually necessary that the form 
of the function E in the transforming equation, and the form of the fimction X under 
the sign of integration, should have a certain connexion. The connexion implied is the 
following. The transforming equation must in general be such, that it may be possible 
by means of it to reduce the differential expression X under the sign of integration to a 
form Y{x, «!, « 2 ? • • which, considered as an explicit function of x and of «„ a.^, . . 
shall be rational with respect to x. This is not a necessity a priori. It is a necessity 
founded in the limitations of the powers of analysis. 
