36 



Sven Wicksell 



Further we have : 

 2a 2 a % = 2 a*[= Z 



1 a t a 6 = 2 « 6 2 = 2 a 2 2 \ 6 4 = — Vs 1 b 2 

 v bi &6 = __ i/ 3 v &i 3 = 1/3 v 2b 5 b,= 2 b 10 2 



V Ci Cg = _ l/ s V = _ l /3 y ß> > v c 2 Cg = 3/ 4 V ^2 = 8/ 4 2 - C 2 



2" c 2 c 4 = — '/« 2 Cj 2 2 c b c 1 = — 2 c 2 



2 Cj c 8 = Vi 2 c 2 



2 c„ c B 4- 2 c 4 e g = 2" c 10 2 — 1 h 2 c 5 2 



2rf 2 tf 4 = 2d 6 2 2d 3 d 7 = 2d> 2d, d d = 2d^ 



2 d, d e = 2 d 2 2 d 1 d s = 2 d 2 2 d 5 d, = 2 d 10 2 



2 d 3 d 5 = 2 d 2 2 d 2 d d = 2 d 2 2d 6 d s = 2 d 10 2 



and all the other 87 sums of products are zero; consequently here only the sum 

 S c 4 Cy need be computed directly, else only sums of quadrats are needed. 



By the equations for the characteristics of the fourth order we finally get: 



2 a\ 



a 2 





2a\a' 10 



= 2 a 2 , 



2V,a' 1( , 



= 2 a 2 , 



2 a' L a' 6 



= 2a'^ 



2b\ 



6' 6 





Zb\b\ h 





2V h b\ 



-lb',, 2 , 



2b' 5 V u 



= Zb\ 2 









2b\b' u 





2b\b\ 



= 2b\ 2 \ 



Zb'sb'u 



= Zb' x 2 



2 b\ 



b' u 





= 2b' 15 \ 



2 b\ 



b' 13 



=^v> 



2b' 10 b' n 



= 2V 2 



13 1 



2b' 10 b' J2 



= Xb\i, 



2b' u b' ia 



= Zb\ 2 



2b\ 



b\ 



= 2b\,\ 



2 b\ b\ 



= 2 y 2 



2b\b\ 0 



— yy 2 



2b\ b' u 



= 2b' 2 



2b\ 



*'u 



= 2b\ 3 \ 



2 b' 2 b\ 



= 2b' 12 \ 



2b' 2 b\ 0 



= 2 b' 2 , 



2b\b' n 



= 2b\* 



2 b' 2 



6'» 



= 2 6' 9 8 , 



2b\ b\ 0 



= 2h' 2 



— 0 15 ' 



2b' 3 b' n 



= Zb' 5 2 , 



2b' 3 b' 12 



= 2b' 2 



2e\c\ = 2 2c' 1(1 2 - { /22c\ 2 - 'hi 2c' 2 , 2 c\ c 10 = - 2 c' n 2 - 2 c 2 -f 'A 2 c 2 

 2c\c\ = h k2c' u 2 — l h2c'*— 'k2c', 2 , 2c' 2 c' 10 = — 2 c' u 2 + ' Ii 2 c 2 — l /i2c' 2 2 



C 1 C 11 = C 13 2 > ^ C 11 C 13 = ^ C 14 



2d\d\ ^ï2d'J — x ^2d! 2 — Vt2<X 2 , 2d' i d\. = 5 A 2 rf' 10 2 — Vs 2 d' 2 — Vs 2d'* 

 2 d\ d' w =-2 d\ 2 - \U 2 d\ 2 + V* 2 d' 4 rf' 15 = 2 ^' 14 2 - V» 2 d\* 



2d' 2 d' i0 =-2d\ [ ?+>/ i 2d\ 2 - V^d 2 2 , 2X^'i5 =2^' 13 2 - 1 /^" , ^' 9 2 



v d' 10 d n 4 /9 ( r,, 2 - 2 d' 7 »), ^ d' 10 rf' 12 = - v» ^' 9 2 - ^ rf' 7 2 ) 



Irf^'^VsIf/V, 2d' 5 d', =Vi2d' u 2 , 2d\d' V2 = — s h2d' 1 2 



I d' 2 d\ x = — 8 /9 2 rf',, 2 , 2 d\ d\ 2 = 79 2 d' 9 2 , 2" «r n d\ 2 = — 2 d\ 2 



2 e\ e' 10 = 2 / 3 2 e' ( . 2 - l h 2 e\ 2 , 2 e\ e' u = Vs 2 e! 2 — Vs 2 e' 9 2 

 y e \ e' 16 = Vs 2 e 2 — Vs 2 e' 7 2 , 2 e' 7 e' 14 = Vs 2 e' 2 - 2 /s 2 e'. 2 

 v e ' g e ' i3 = i/ 3 2 e' 7 2 — Vs 2 e',f, 2 e' 2 e' 10 = 2 /s 2 c' 4 2 — Vs 2 é 2 



^ e 10 e 10 ~ — e 13 s ^ e :i 2 ' ^ e 10 e 12 = ^ e \i V3 2 e 7 2 



2e\e' 2 = 2e' 2 , 2e\e\ = 2e' 2 , 2e\e' n = 2 h2e' 2 , 2 e\ e' t2 = 2 /s e 2 



^ e II e 12 = 2 e 15 2 - 



