38 Mr. C. J. Hargreave's Analytical Researches 



/" dx 

 -, ; which, although 



not used as commonly as logarithmic or trigonometrical func- 

 tions, has received a distinctive name (logarithm-integral) and 

 symbol (li.2?), and has been tabulated by Soldner from to '99 

 and from I'l to 1280. The table is given in Mr. De Mor- 

 gan's DifF. Cal., pp. 662, 663. It will be useful to note the 

 following properties : 



Px'^dx ,.- ^,,, 



y„b^ =''("')' 



(except when m= — 1, in which case 

 Jj^ =logloga;,) y ^ =li(6*), jY\xdx=x\\x--Yi{a^). 



The following researches do not imply or assume any know- 

 ledge of the observed values or deduced properties of prime 

 numbers, except so far as is expressly mentioned ; but I have 

 not on this account thought it necessary to employ symbols in 

 lieu of well-known primes (such as 2, 3, 5, 7 . .), where the use 

 of the numbers places the expressions in a more familiar form. 

 The symbol p is used to denote a prime number in general ; 

 and accents or suffixes are attached when necessary for distinc- 

 tion. 



Prop. 1. If 



«"" "pi 2^ 3« 4^ 



where 1, 2, 3, 4 ... denote the series of natural numbers to 

 infinity, then 



where 2, 3, 5... denote the series o^ prime numbers to infinity. 



If we take the natural numbers beginning with 2, and strike 

 out from the series the first prime (2) and all its multiples, 

 then the second prime (3) and all its multiples, and so on 

 through the series of prime numbers, we shall have struck out 

 all the numbers in the following manner : the primes will have 

 been struck out, once; the composites of two primes, twice; 

 the composites of three primes, three times ; generally the 

 composites of m primes^ m times. 



If we now go through the series again, and restore 2.3 and 

 all its multiples, 2.5 and all its multiples, 3.5 and all its mul- 

 tiples, and generally PxP'i. ^^^^ ^^^ ^^s multiples, going through 

 the series of primes, we shall have restored the composites of 

 2 primes, once ; the composites of 3 primes, three times ; the 

 composites of 4 primes, six times ; and generally the compo- 



