CEETAIN APPLICATIONS TO THE THEOEY OF DEFINITE INTEGEALS. 793 
those limits, and therefore, by consistency of interpretation, f{a) as vanishing vrhenever 
<r transcends the same limits. 
40. I shall not enter into any discussion of the above solution, but shall briefly point 
out in what way it may be generalized by the theorem of definite integration of Art. 32. 
It is evident that if we change in (6.) 
into 
into Xo 
-fta 
^2 H'fn 
See. 
See. 
See. 
or into any of the remarkable forms thence derived. Art. 36, leaving dx^dx ^ . . dx„ un* 
changed, the actual value of the multiple integral will be unaltered. Thus, as a par- 
ticular illustration, if we suppose 
V =^^^dxdi/dz 
the integrations being limited only by the condition 
we should find 
where 
•^0 
4i ’ 
provided that/’((r) = 0 when tr does not fall within the limits 0 and 1. 
41. Example 2nd. Let 
V=J. . dx,dx, . . dx„^ ^ 
the integration extending to all values of the variables which satisfy the condition 
+^) < 1 ’ 
Here, after reductions similar to those which have been exemplified in the preceding 
problem, but more complicated, we find 

r(*)Jo + ^ 
5 I 4 
MDCCCLVII. 
