32 



Sven Wicksell 



At last we write down the equations for the direction cosines of the vertices. 

 In accordance with the theory of linear substitutions they are: 



I K" *lKl +Pi £ 21 + *1 8 81 = ° 



Direction cos. of o x I p Y s u -f [y 1 — £ 2 i + 9i £ 3i = ^ 

 ' r i s u + Qi £ 2i + *i) hi = 0 



I *l) £ 12 + i»l £ 22 + »1 £ 32 = 0 



(63) Direction cos. of a 2 Pl e u + (y t — «,) s 22 + g, s 32 = 0 



I ?"l £ 12 + Ï1 £ 22 + (»1— *l) £ 32 = 0 



I («1— S s) £ 13 + #1 £ 23 + »1 £ 33 = 0 



Direction cos. of a 3 I p l e 13 + (?/ x — s 3 ) s 23 4- Qi s 33 — 0 



I r i £ 13 + ?1 £ 23 + K— S s) £ 33 = 0 • 



And for the declination and right ascension of the vertices we have 

 Vertex I Vertex II Vertex III 



cos 6 X cos = £ n cos § 2 cos a s = s 12 cos S 3 cos a 3 = e 13 



cos § t sin oq = î 2I cos 8 2 sin a 2 = e 22 cos § 3 sin a 3 = s 23 



Sin8 l = £ 3.- SniS 2 = £ 32- Sin§ 3 = £ 33- 



15. Using the equations (41) or (43) we obtain as equations of condition for 

 the characteristics of the third order, from each square a system of the following- 

 form : 

 (64) 



^30 = «1 #"300 + «2 B "u30 + »3 B "oOS + a 4 B "â10+«S #"«01+ fl 6 B "l20 + «7 ß "o21 + a « B "l02 + fl 9 £ "oi2 + a iO B " X 

 ^2! », #"300+ &2 #"030+ &3 B"00.+ &4 5 " 21o + &5 #" 2 0, + \ 120+h #"o21 + B" w2 + \ B" Qn + b 10 B'\ 



B 12 7 r i #"3ü0+ C 2 5 "ü3ü+ C 3 #"003+ C 4 #"210+ C 5 5 "201+ C 6 £" 120 + C 7 ß "o 2 i+ C « B "l02 + C » - B "o'l2+ «10 B 1 



012 + ^10-^ 1 





w 



r here the coefficients are given by: 



(observe 



T31 



= 0) 













a , 



= T 3 i 1 





&! = 3? 2 j 



1 Yl2 





c i 



= 3 Ti, 



Tia 









= 7 8 12 



«2 









1 7 2 2 





C 2 



= 3t 21 



72 2 







^2 



= Yaa 



a 3 



= 7 3 3 i 



— 0 



^3 = 3 Ï3 



! 7 3 2 = 0 





C 8 



= 3 T31 



7 3 2 = 



= 0 





d 8 



= 7s 2 



a 4 



= Ti 1 







i 7 1 2 7 2 1 +7i, 



72 2 



C 4 



= 2Yh 



Yi 2 T 2 



2+YÏ 



2 7 2 1 





= Y 1 2 Y 2 2 



«5 



= Ti 1 



Y 31 =0 



*5 =27, 



1 Tu Tai +TÎi 



7 3 2 



c 5 



= 2 Tll 



7 1 2 7 3 2 ~l~ 7 1 



2 73 1 



d 5 



= Yi 2 Y3 2 



a 6 



= Ï11 



T21 



& 6 =2 T] 



1 7 21 Ï22 ~t~7l2 



Yli 



c 6 



= 2t.i 



Y22Y, 



2+Y1 



l 7 2 2 2 



d e 



= Y 1 2 Y 2 2 



a 7 



= Tai 



T 31 =0 



6 7 =2 Ï2 



1 7 22 Ï3 1 +T|, 



Y 32 



C 7 



= 2y 21 



Y22Y3 



2+72 



2 7 3 1 



à, 



" 7<J 2 73 2 



«8 



= Ti 1 



Y 3 i = 0 



^8 = 2t, 



iT 3 iT 32 +Ti2 



r 3l = o 



c s 



= '2Yi2 



Tai Ta 



2+7, 



1 7 3 2 



d 8 



= YiaY 8 2 



«9 



- T21 



T'.i-O 



& 9 = 2 Ï2 



1 73 1 73 2 H" T 2 2 





c 9 



= 2 Ï2 2 



73 1 73 



2+Ta 



1 7 3 2 



d 9 



= 7 2 2 Y 3 2 



a, 



0 = Y 1 1 



Y^i 7 3 1 = 



0 &io= T, 



1 72 1 73 2 





Cl. 



)= Yi, 



72 2 73 



2 





d l{ 



> = Yi 2 Y 2 2 Y32 









+ Ti 



iTazTai 







+ 7 12 



72 1 73 2 















+ 7, 



■ Tai Tai 







+ Y, 2 



72 2 73 











