646 



REPORT — 1900. 



When the boundary is the infinite plane with hemispherical bowl, we con- 

 sider our Physical Space as driven by 



o<e<^' 



and require three revolutions of the radius vector to complete our Multiple Space. 

 The poles will now be at ( ± $'), ( ± (Stt + ^0)- 



The general solution (of period 2n7r) required for such problems is given by 



M = 



J^ Vcos hp — cos 6. cos hp' — cos 6' 



smA- 

 n 



wn-A/2 

 where 



^^7^ a a' 



^co8 ha - cos ha' Cos A - - cos 



n n 



,da, 



cos ha' = cos hp COS hp' — ain hp sin hp' cos ((f> — <f)'). 



6. Determination of Successive High Primes. By Lieutenant- Colonel 

 Allan Cunningham, R.E., and H. J. Woodall, A.R.C.Sc. 



A general method is described of determining, in a compendious manner, the 

 luhole of the primes within a given range. A table is given showing the lowest 

 factors of all the numbers between (2^^ T 1020), i.e. between 16,776,196 and 

 16,778,236, which brings them all withia the power of the existing large factor- 

 tables. This determines the whole of the Hiffh Primes (117 in number) within the 

 above range, as in table below : — 



List of 117 High Primes between (2** T 1020). 



7. On the Construction of Magic Squares. By Dr. J. Willis. 



8. The Asyzygetic and Perpetuant Covariants of Systems of Binary 

 Quantics. By Major P. A. MacMahon, F.R.S. 



The perpetuant theory connected with a single binary quantic has been com- 

 pleted as a consequence of the fact that all the asyzygetic seminvariants of given 

 order are known and easily re presentable. The object of the present paper is to 

 extend this knowledge to the seminvariant forms connected with a system of 

 binary quantics. The method of Stroh is then applicable, and the complete theory- 

 can be established without difficulty. 



