E Q U 



In like manner, B = K2; so that the equation becomes, 



y = (lo. 54'.2) sin. x -f (1'.2) sin. 2 jr. 

 This is nearly the equation of the centre in the earth's orbit. 



In this \vay all the elements of any of the planetary orbits may be de 

 termined simultaneously, or corrected if they are already nearly known. 

 In the construction of Astronomical Tables, the number of equations 

 combined has amounted to many hundreds. 



In the example above, no method was to be followed, but that of di- 

 viding the original equations into two parcels or groups, from the sums 

 of which the new equations were to be deduced. But when it happens 

 in the given equations, that the terms involving the same unknown quan- 

 tity have different signs, the best way is to order all the equations so that 

 one of the unknown quantities, as A, shall have the same sign through- 

 out j and then to add them together, for the first of the derivative equa- 

 tions. Let the same be done with B, C, &c. whatever be the number of 

 the quantities sought. Thus, each of the unknown quantities will occur 

 in one of the equations, with the greatest possible coefficient; and the 

 coefficients of the same unknown quantity, in the different equations, will 

 become by that means as unequal as they can be rendered, which con- 

 tributes to make the divisor by which that quantity is to be found, as 

 large, and itself of course, as accurate as the case will admit of. 



Ex. Let the equations be 



3 x+y 2z = o 

 5 3 .r 2 y + 5 * o 

 21 4.r # 4# o 

 14-h.r 3 # 3z = o 

 changing the signs of the last equation, and adding, 



15 9x + y + 2z = o 



similarly for #,37 5 or 1 y = o 



forz, 33 # y Uz = o 



From these equations x = 2.486 

 y = 3.517 

 z = 1.928 



Second Method. 



Let m + ax + by -f- cz + &c. = o, 

 m' + a' x -f b' y -f- c' z -f &c. = o, 

 m" -f- a"x -f b"y + c" z + &c. o, 



&c. ,..,, , ,., ~ o r 



104 



