2] RELATION TO THE MOTION OF THE NODE OP THE MOON'S ORBIT. 97 



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Or if, as before, we put 



2 =v = g Lf?? ==K . = 9! l^ =a = r /! I^ = & = 2! etc 



k y 2 ' AT ' 4 ' 2 Tc 4 ' 2 & 2 4 ' 



the equations become 



= . . . [(y 2) 2 K"~\ c_., ac_ l bc cc^ dc., ... 



= ... ac_ 2 +[(yl)- K-~\c_ 1 ac () bc l cc., ... 



= ... bc_ 2 ac_ l + [y 2 /c 2 ] c a^ 6c 2 ... 



= ... cc_ 2 -6c_! -ac + [(y+l) 2 -/c 2 ]c 1 ac, ... 



= ... cfc_ 2 cc_ I -6c ctCj + [(y + 2)--/c 2 ]c, - ... 



Divide each of these equations by the coefficient of the term in the 

 diagonal line, and form as before the determinant that gives the value of y. 



With a view to determining which terms it is necessary to consider, 

 write down all the elements of the minors up to those of four rows and 

 columns, together with the powers of , I), c, &c. which they involve. In the 

 notation below the position of each figure represents the row it is drawn 

 from, and its value represents the column, thus 123 represents the diagonal 

 element for three rows and columns. The sign of each term is also given. 



Two Three Four Four 



(1) 12 1 (3) 123 1 (9) 1234 1 (21) 1423 a~b 



(2) -21 a' (4) -213 a 2 (10) -2134 a 2 (22) -2413 a*6 s 



(5) -132 a 2 (11) -1324 a 2 (23) -1432 V 



(6) 231 a>l> (12) 2314 orb (24) 2431 abc 



(7) 312 a 2 6 (13) 3124 a*b (25) 3412 & 4 



(8) -321 6 2 (14) -3214 fc 2 (26) -3421 alfc 



(15) -1243 a 2 (27) -4123 a 3 c 



(16) 2143 a 4 (28) 4213 abc 



(17) 1342 a 2 6 (29) 4132 abc 



(18) -2341 a 3 c (30) -4231 c 2 



(19) -3142 tfV (31) -4312 aVc 



(20) 3241 abc (82) 4321 aV 

 A. [I. 13 



