.30 Sven Wicksell 



Now we know that the determinant A is invariant when changing the direction 

 of the axes of coordinates. Thus 



Further we have, as 



and writing 



1 



m = F" - 



él = N - ?l = N - ?1 = N 



g ^02' g A '20> g ""111 



B"C"—D" a „ o A"C"~- E" 2 „ . a"B"—F" 2 a , 



A" 1 ' ;/1 A" 1 ' 1 A" 



D"E"—G"F" „ . E"F"—A"D" ft . F"D"—B"E" 



A" 1 ' « A" 1 ' 1 A" 1 ' 



we obtain for asj , y t , ^, p x , ^ t , >'j the linear equations 



V^O« = X l Ï12 2 + yi T22 2 + «1 Ï32 2 + Ï12Ï22 + 2 2l Ï22Ï32 + 2 >'l Yl2 Y 8 2 



(57) V N 20 — x x -i u 2 + y x t 21 - + ^ Ï31 2 + 2 Pl TllT|1 + 2 ^ T 21 T 81 + 2r, Tn t 31 



V ^11 = Tu T12 + V\ Ï21 Ï22 + *i T31 Y32 (Tu Ï22 + Y12 T21) H" 2i (Tu T32 + T22 T31) 



+ *1 (TsiTl2 + T 8 2Tll)- 



Inserting the expressions (55), and as y 31 = 0 , we get 



v o2 9' — (i— 2') = »1 «i + s + *i a 3 + Pi a 4 + 2i a 5 4- «g 



(5 s ) v 20 g' - V = ^ ß 4 + Vi ß 2 + i>, ß 4 



v u 2' -- #0 y 0 (!— 2') = Ä 'i Ti + 2/1 T 2 + Pi T 4 + 2j T 5 + r i Ye » 



where 



*l = Yl2 2 a 2=Ï22 2 a 3=Ï32 2 «4 = 2 Tl2 T 22 «5 = 2 T 2 2 Y 32 K 6 = 2 Yl2 Y M 



ßl — Vil 2 f 3 2 = Ï21 2 $4 = 2 Yll Y21 



Yi — Tu Y12 Y2 = T21 Y22 '1 4 — (Yh Y22 "H T12 T21) Ys === T21Y32 Ye— Yn Ys2 



On account of y 81 = 0 we have besides 



Yi = — Y 2 1 Y s = + Yi8 > Y 6 = — Y 23 • 

 Letting q remain undetermined we get normal equations having the left 

 membrum of the form 



«1 2' — \ (1 — 0.') • 



Accordingly the quantities , y t , e 1} p lf q x , r t are expressed in the same 



way. 



To more clearly point out the genesis of the equations (57) we remark that 

 according to (10) we have 



