GENERAL THEOREMS ON MULTIPLE 

 INTEGRALS-. 



IN Liouville's theorem for the reduction of a certain class 

 of definite multiple integrals, the integrations comprise all 

 positive values of the variables which do not transgress a 

 limiting inequality, which either is of, or may easily be re- 

 duced to, a linear form. Take for illustration the case of two 

 variables, and let mx + ny < h be the limiting inequality in 

 question, m, n and h being positive. Then, geometrically, 

 mx + ny h is the equation of a straight line which forms the 

 base of a triangle of which the intercepts of the positive half 

 axes of co-ordinates are the sides, and our integration extends 

 over the whole surface of this triangle. A similar interpreta- 

 tion may of course be given in the case of three variables. 

 But to return to that of two. Let mx -\-ny-Ti cut the axis 

 of x in the point M and that of y in the point N: conceive 

 another straight line mx + ny h' ; m', n, h' being also all 

 positive ; and let it cut the axes in M' , N' respectively. Let 



us suppose for distinctness that , is greater than . Then, 



if the value of T , be intermediate between those of the two 

 h, 



fractions 7 and , it will be easily seen that the two lines 



must intersect in some point A, lying in the positive quad- 

 rant of co-ordinates, and that we shall have a quadrilateral 

 OMAN', (0 being the origin of co-ordinates,) formed by the 

 axes and by the two bounding lines. If now we integrate 

 any function of x and y for all positive values of the variables 



* Cambridge and Dublin Mathematical Journal, Vol. I. p. r, 1846. 



