165 



L'7' <''<''■ 1 I ^- [r/''V^ I >i ^ |.'/('VM S + . . + [.'?,• c///»,] + [qj y/]' =: 0, 



IV. To delenniiie the weights of tlio directions x, y, z, . . ., we 

 again begin by shifting the origin (by translation) from 0' to the 

 point P, satisfying the normal equations and Wj = 0. 



Calling l\ the i)0tential in P, V — [T^^zW the difference of 

 potential relatively to F, §',»/, s', ■• . the coordinates with respect 

 to P, and putting finally 



cciê' + ^nï + y/ ?' + ..= W' , a/ -e + ,?/ ,f + yy $' + ...= tt:,-' 

 we find 



2fr=b;F;-|-2[^,TF/J'. 



This equation represents the set of equipotential spaces 52. U'=0 

 furnishes the hyperellipsoid ii„ tonching (or Rj in 7^. 



Now those points must be found at which the force can only be 

 resolved into an (inactive) com[)onent perpendicular to li and a 

 component parallel to the .'v;-axis. 



For such a point we have 



dU' 

 dU' 



or' 



Of 



[piVi' ai] - [qja/y =^ - f)y«/j' + g^^\ 



» i . » ■ . i . k t . I 



or putting 



Kj — ']j ^ «/' 



[/'.«.• ■^^/'j + UWl'^/.-e', \pii3^ ^T'] + [«;,Vl-0, [;.vy, F/J + [.;y;j'=_-0,rtc. 

 whence 



[Pia;'] S' + [p/«/^/] li' + [p/«r//] C -f . . . + [sj-«/ 1 ' — (/,-=', 



[pir ^<i]ê' + [pin^ii] V + f /vy;^] $' 4- • • . + [-vy/J' = o, 

 or 



Proceedings Royal Acad. Amsterdam. Vol. XVU. 



