TRAXSACTIOXS OF SECTION A. 



553 



8. Determination of Successive High Primes. [Second Paper.) 

 By Lt.-Col. Allan Cunningham, R.E., and H. J. Woodall, A.Ii.C.Sc. 



A general method was previously explained of determining, in a compendious 

 manner, the tchole of the primes within a given range. Tables have now been 

 prepared showing the lowest factors ( ><'5) of all the numbers between (2-' t 1020), 

 I.e., between 33,553,412 and 33,555,452, thus bringing them all within the power 

 of the existing large factor- tables. Hereby are detected the w/iole of the High 

 Primes (128 in number) within that range, and also the ic/iole of the Secondary 

 High Primes (45 in number) contained as factors of the numbers within that 

 range. [The u-hole of the work required has been done by each of the joint 

 authors independently.] 



There is a long sequence of 73 composite numbers between 33,554,393 and 

 33,554,467, and one of 51 composites between 11,184,889 and 11,184,941. 



List of ^5 High Primes between ^ {2-'' T 1080). 



9. The Equation of Secular Inequalities. 

 By T. J. I' A. Bromwich, St. John's College, Camhridge. 



The theory of the mean motion of the perihelion and node of a planet's orbit 

 was proved by Laplace to depend on a certain determinantal equation of degree 

 equal to the number of planets considered. A paper has recently been published 

 by C. V. L. Charlier (' Ofversigt af kongl. Vet.-Akad. Forhandlingar,' Stockholm, 

 1900, p. 1083) ill which he considers the question of equal roots in this equation ; 

 the case of equal roots was regarded by Laplace as unstable. Charlier remarks 

 that Weierstrass had proved (' Berliner Monatsberichte,' 1858, p. 207, or * Ges. 

 AVerke,' Bd. i. p. 233) that the equation (of the same form) which appears in the 

 theory of small oscillations about a position of equilibrium does not lead to 

 instability if equal roots are present ; but apparently he regards Weierstrass's 

 investigation as not sufficient to apply in the more general problem of astronomy. 

 Charlier then considers the astronomical case, in Weierstrass's way, supposing that 

 equal roots do appear in Laplace's equation ; but the astronomical case may be 



