1877.] Integration between the limits of Summation. 47 



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

 system p^ 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 a, 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. 



(3) Mr J. W. L. Glaishee, M.A., F.R.S. Preliminary 

 account of an enumeration of the primes in Burckhardt's tables (1 to 

 3,000,000.) 



The present paper is a continuation of that published in 

 pp. 17 — 23 of this volume, and relates to the enumeration of the 

 primes between 1 and 3,000,000, the enumeration being made 

 from Burckhardt's Tables des diviseurs (Paris, 1814 — 1817). The 

 work was continued regularly upon the printed forms described on 

 p. 20, and was performed with such care that on comparing the 

 new calculation with the old duplicate calculation, the former was 

 found to be almost wholly free from error : in fact, two millions 

 were quite correct, and only two errors were found in the third 

 million. As for the old calculation, the first million was free from 

 error ^ ; but several errors were found in the second and third 

 millions. Considering that the new calculation was performed by 

 a very careful computer, who had had no connexion with the earlier 

 work, and that the old calculation was itself the result of at least 

 two independent enumerations, I feel very little doubt that the 

 numbers given in the present paper may be depended upon with 

 confidence. 



1 The old enumeration was made from Chernac's Cribrum Arithmeticum (1811) ; 

 the new one from Burckhardt's tables, and all the discrepancies between the two 

 were due to errors in Chernac which are noted by Burckhardt on the first page of 

 the preface to his First Million (1817). 



