BETWEEN LIMITS BY SUMMATION. 611 



The solution of these equations gives two systems of values : 



First System. Second System. 



^, = 0-969773, jp 2 = 2-638430. 

 fc= 1-119224, ^ = 0-602526. 



r, = 0-652817, r, = 0-855044. 



0,= -2-856446, 2 = -7'999462. 



P,= 1-106789, P 2 = 0-000513. 



Q,= 0-032304, Q,= 1-238514. 



12,= 0-478508, R,= 0-094777. 



In the first system q l is greater than unity, and in the second system 

 p t is greater than unity, so that in either case one set of the values of u 

 corresponds to values of the variables outside of the limits of integration. 



This, of course, renders the method useless in determining the integral 

 from the measured values of the quantity u, as when we wish to determine 

 the weight of a brick from the specific gravities of samples taken from 27 

 selected places in the brick, for we are directed by the method to take some 

 of the samples from places outside the brick. 



But this is not the case contemplated in the mathematical enunciation. 

 All that we have proved is that if u be a function of x, y, z of not more 

 than seven dimensions, our method will lead to a correct value, and of course 

 we can determine the value of such a function for any values of the variables, 

 whether they lie within the limits of integration or not. 



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