[ 412 ] 



— Sai>, •.• a^ = — 6ba — b' and a — — '^6 + zb V z ; io that either 

 b, or a, will be cmbarraffed with a furd. 



Let the prime number whofe logarithm is fought by the ap- 

 plication of the above produds of four dimenfions, be denoted 

 by X + 2, if the number next greater is meafured by 3 ; by 

 a; + 3, if the number next lefs be meafured by 3 — and then 

 all the other fadors will be compofite. For in the firft cafe, 

 X — I will be meafured by 2 ; x, by 3 ; x + 3, by both 2 and 

 3 ; X + 5, by 2 ; and ^ + 6, by 3 : In the other cafe, x — i, will 

 be meafured by 3 ; ;c, by 2 ; ;v + 2, by both 2 and 3 ; «■ + 5, by 

 3 ; and X + 6, by 2. 



I F the prime number, whofe logarithm is fought, were de- 

 noted by any other fador than x + 2, or at + 3, other divifors, 

 befides 2 and 3, fhould come into confideration ; for in any cafe 

 there would be at leaft another fador, which, neither 2, nor 3, 

 would meafure. 



Using then the notation direded above, the ratio of ;v* x .v + 5]= 



-«. ;<■ + 5 ^ 



to, A- — I X X + 2 X X + 3 X «• + 6, of 



X — 1. ^■ + 3. x + 6 

 : vV+ 2, 



