The general Characteristics of the freqnencyfunction of stellar movements 35 



Similarly as in the case of the characteristics of the third order we now write 

 the left membra thus 



(67) 



V B i0 = a\ % q 6 + a' 17 ( 1 - </ s ) q'* + a' 18 (2 - q'- 2q'* + 2 ' 6 ) q + a' 18 ?' 2 + a' 20 (6 - 1 2 2 ' + 3 2 ' 2 + 4 2 ' 3 - 2 ' 6 ) 



V # 04 = &' 16 ?' 0 + h n 0 - ?' 8 ) '/ 3 + 6'i8 (2 - 4- 2<z' 2 + 2' 5 ) '/ + b' 19 q" 2 + fe' 20 (6 - 12 2 ' + 3 3 ' 2 + 4 2 ' 3 - 



V # 81 c' 16 2 ' 6 + c' 17 (1 - 2 ' 3 ) q" + c' 18 (2 - ?' - 2 2 ' 2 + 2 ' 5 ) 2 ' + c' 19 2 ' 2 + c' 20 (6 - 12 2 '4- 3 2 '» + 4g' 3 - 2 ' 6 ) 



V B u = d' w q'« + d' u {\ - q'*) q' s + rf' ls (2 - 2 '- 2 2 ' 2 + 2 ' 5 ) 2 '+ «T 19 2 '* + <T 20 (6 - 12 2 '4- 3 2 ' 2 + 4 2 ' 3 - 2 '«) 



V 5 22 = e' 16 2 '« + e' 17 (1 - 2 ' 3 ) 2 ' 3 + e ' 18 (2 - 5 - 2 2 ' 2 + 2 ' 5 ) j'+ «'„ 2 '* + e ' 20 (6 - 12 2 '+ 3,V 2 + 4 2 ' 3 - 2 '«) 



and we find from (55**), (30*) and (17) 



«'ie — + v 4o : 24 



ft 'l6 = 



+ 



v • 24 



C'l6 = + V 31 : 6 



a' u = — v 80 x 0 : 6 







V 03 #0 : 6 



c'n = --(3v 21 x 0 + v 30 «/ 0 ) : 6 



ö 18 = -j- v 20 x 0 : 4 





+ 





C 'l8 = + ( V U V + V 20*oi/o) 



a 19 = — v 2o 2 : 8 



ft 'l9 = 





V2 2 : 8 



c ' = v v -2 



0 19 20 11 " 



a' 2(1 = — V : 24 



&' 20 = 





y 0 * : 24 



c' 20 = — x 0 3 y 0 : (1 



d 'ie = + v i3 : 6 







e 'i6 = 



+ v 22 : 4 



d 'n = — ( 3v i2 Va + 



v 03 oî 0 ) : 6 





e' 17 = 



+ v 21 ^„):2 



rf 'i8 = + ( v n y 0 2 + 



"<>« - r o y 0 ) : 2 





e 'l8 = 



+ (4v u x- 0 y 0 + v 20 »/ n 2 + v 02 ar 0 a ):4 



^ 1!) == V 02 V ll • 2 







e'19 = 



- (2v u 2 + v 20 v 02 ):4 









e ' 2 o = 



— x 0 2 y, 2 : 4. 



Having computed and solved the normal equations we will have as solutions 

 the numerical values of the Hiß in the following expressions: 



(67*) V = H% 2 '« + m> (l- 2 ' 8 ) 2 ' 3 + Äg (2- 2 '- 2 2 ' 2 + 2 ' 5 ) 2 ' + Äg 2 ' 2 + 



+ i/§((3-12 ? ' + 3 ? ' 2 +4 2 ' 3 - 2 ' 6 ). 



17. Now it would seem to be very laborious to undertake to compute and 

 solve normal equations with fifteen unknowns especially as the unknowns come 

 forth as algebraical polynoms of q . However, a multitude of circumstances are at 

 hand to facilitate the work in an astounding way. I will here mention a few that 

 depend on the symmetrical distribution of the squares with regard to the equator. 



When computing the normal equations we have to form sums of products of 

 the coefficients, for instance X a'i a'j , over all the squares. Now it may be shown that 

 in all cases except five, the sums of products are zero or expressible in the sums of the 

 quadrats of some of the other coefficients. Indeed, it may easily be verified that 



- a x a 2 = l /i 1 a, 2 , Ï oq a g = V* - - a 2 a 3 = S ot 5 2 



Sß 1 ß 2 = 1 AS ß 4 » £ ïi Y, --S V. 



All the other 23 sums of products are zero and here consequently only the 

 sums of the quadrats need be computed. 



