122 M. G. H. Dahlander on a New Species of Figures 



If we now take ^ — n.=E, S = m. we find the differential 

 27r P 'f ft 



equation for the surfaces de niveau to be 

 {-"nrI Vl: ^-'' (X + VI+J?)] + (m-l)^l+^[x'v / iTx r ' 



^■{-^[MHv/IW,-^] 



+ s7TO+^}^ =0 (7) 



But for the surface de niveau which forms the inner bound- 

 ing surface \/a; 2 + y' 2 + z' 2 =r' 2 , and we have therefore for it 



"Ti^rJ + 2 ( m - 1 ) ^W^[ 1 ' ( v +v / TTxS) 



-vm<] + i} M < 8 > 



But for the corresponding values of x, y, z we have the relation 



xdx + ydy + zdz=0 (9) 



By comparing the equations (8) and (9), therefore, wc get 



+ 3( OT -]Vr^ M v +v / CT ) (10) 



This relation must exist between the several quantities in 

 order that the inner bounding surface may be a sphere. If we 

 now take 



+2(„-i)*^+*'[,. (V+ ^n.x«)-- 7I E 5 -J=r, (ii) 



