

DETERMINATION OF ELLIPTIC ORBITS. 119 



If we adopt such a unit of time that the acceleration due to the 

 sun's action is unity at a unit's distance, and denote the vectors* 

 drawn from the sun to the body in its three positions by 9^, 9? 2 , $H 3 , 

 and the lengths of these vectors (the heliocentric distances) by r lt 

 r 2 , r 3 , the accelerations corresponding to the three positions will be 



<v\ <x\ 03 



represented by -- 1, -- 1, -- 1. Now the motion between the 



r i r z r s 

 positions considered may be expressed with a high degree of accuracy 



by an equation of the form 



having five vector constants. The actual motion rigorously satisfies 

 six conditions, viz., if we write T 3 for the interval of time between the 



* Vectors, or directed quantities, will be represented in this paper by German capitals. 

 The following notations will be used in connection with them : 



The sign = denotes identity in direction as well as length. 



The sign + denotes geometrical addition, or what is called composition in mechanics. 



The sign - denotes reversal of direction, or composition after reversal. 



The notation 31.53 denotes the product of the lengths of the vectors and the cosine of 

 the angle which they include. It will be called the direct product of 51 and 33. If 

 ar, y, z are the rectangular components of 51, and a;', y', z' those of 33, 



51 . 51 may be written 5l 2 and called the square of 51. 



The notation 51x33 will be used to denote a vector of which the length is the product 

 of the lengths of 51 and 33 and the sine of the angle which they include. Its direction 

 is perpendicular to 51 and 33, and on that side on which a rotation from 51 to 33 

 appears counter-clockwise. It will be called the skew product of 51 and 33. If the 

 rectangular components of 51 and 33 are x, y, z, and x', y', z', those of 51x33 will be 



yz' - zy', zx' - xz', xy' - yx'. 



The notation (5133(5) denotes the volume of the parallelepiped of which three edges are 

 obtained by laying off the vectors 51, 33, and (5 from any same point, which volume is to 

 be taken positively or negatively, according as the vector (5 falls on the side of the plane 

 containing 51 and 33, on which a rotation ifrom 51 to 33 appears counter-clockwise, or on 

 the other side. If the rectangular components of 51, 33, and (5 are x, y, z ; x', y', z' ; 

 and x", y", z", 



x y z 



x' y' z' 

 x" y" z 



It follows, from the above definitions, that for any vectors 51, 33, and (5 



51.33=33.51, 51x33= -33x51, 



(5133(5) = (33(551) = ((55133) = - (51(533) = - ((53351) = - (3351(5), 

 and (5l33(5) = 5U33x(5)=33.((5x5l) = (5. (51x33) ; 



also that 51 . 33, 51 x 33, are distributive functions of 51 and 33, and (5133(5) a distributive 

 function of 51, 33, and (5, for example, that if 51 = 8 +WI, 



5l.33=2.33 + 2R.33, 51x33=2x33 + 9^x33, (5133(5) = (33(5) + TO<5), 

 and so for 33 and (5. 



The notation (5133(5) is identical with that of Lagrange in the Mtcanique Analytique, 

 except that there its use is limited to unit vectors. The signification of 51x33 is closely 

 related to, but not identical with, that of the notation [r-^r^ commonly used to denote 

 the double area of a triangle determined by two positions in an orbit. 



(5133(5) = 



