86 Mr Glaisher, Tables of the number of numbers, Sc. [Jan. 28, 



form, for its form can only slightly differ from that of an oval of 

 intensity k = 0. 



Thus the curve near A has its concavity inwards. 



In order that the curve may turn so as to become closed it is 

 clear that the concavity of the end BD must be outwards. 



Thus a point of inflexion must exist at B somewhere between 

 A and D. 



Similarly there is by the symmetry of the curves a point of 

 inflexion at G. 



Thus on a curve near to an oval of intensity there are two 

 points of inflexion. 



The equation for finding these points is not difficult to obtain. 

 But it will be found to be so exceedingly complex that it is prac- 

 tically useless, and on this account it is not given. 



(5) Tables of the number of numbers not greater than a given number 

 and prime to it, and of the number and sum of the divisors 

 of a number, with the corresponding inverse tables, up to 3000. 

 By J. W. L. Giaisher, M.A., F.R.S. 



[Abstract] 



Denoting by </> (n) the number of numbers not greater than n 

 and prime to it, by v in) the number of divisors of n, and by cr (n) 

 the sum of the divisors of n, unity and n itself being included, 

 the tables contained in the present paper are as follows : 



Table I. The complete resolution of n into factors and the 

 ■values of </> (n), v (n), and cr (n) for all values of n up to n = 3000. 



Table II. The values of n corresponding to <f>(n) as argu- 

 ment. 



Table III. The values of n corresponding to v(n) as argu- 

 ment. 



Table IV. The values of n corresponding to cr(n) as argu- 

 ment. 



Tables II.-IV. are inverse to Table I. and extend also to 

 n = 3000. 



An introduction containing a collection of formulas relating 

 to the functions </> (n), v (n) and cr (n) is prefixed to the paper. 



The paper is in course of publication in the Transactions of 

 the Society. 



