June 8, 1917] 



SCIENCE 



575 



then the solution is accomplished only by 

 successive approximations or trials in the 

 course of which, however, higher terms of 

 the series in the ratios of the triangles may 

 be taken into account. It is customary to 

 assume as a first approximation that the 

 sum of the first and third radii vectores is 

 equal to 2 astronomical units. Convergence 

 of the approximations has been facilitated 

 by Oppolzer by differential relations which 

 give the correction to be applied to the 

 initial value of ri + rm so that it may agree 

 with the value derived at the end of the 

 trial. In the course of ordinary trials 

 without the use of differential relations the 

 final values of the distances of one trial 

 form the initial values in the next trial. 

 In the method of differential corrections 

 such corrections to the initial value of one 

 trial are derived differentially from the 

 differences between the initial and final 

 values in the same trial as will produce an 

 agreement of the initial and final values in 

 the next trial. The number of approxima- 

 tions required by the ordinary trials is in 

 general far in excess of that required by 

 the method of differential correction. 



Gauss's method as formulated by Op- 

 polzer may be started from the equation 

 Ax + By + Oc = 0, which expresses that 

 the body moves in a plane through the sun, 

 X, y, z being the heliocentric rectangular 

 coordinates referred to the sun. When this 

 equation is written out for each of the three 

 places and when the eliminant of the three 

 equations is written down in the form of a 

 determinant this determinant may be devel- 

 oped either in terms of co-factors of the x, 

 or the y, or the z. For instance, in co- 

 factors of X we have 



Xiirjiiz 



■ 2/iii2ii) - XniyiZiii — 2/mZi) 



+ a;iii(?/i2n ■ 



ViiZi) =0. 



out by one of these areas the two resulting 

 coefficients represent the ratios of the pro- 

 jected triangles and since the triangles are 

 projected on the same plane these ratios 

 are the same as the ratios of the triangles 

 themselves. As stated before, instead of 

 developing the determinant by co-factors of 

 X it may also be developed by co-factors of 

 y and z. "We thereby obtain the same equa- 

 tion written in two additional forms. 

 Every equation is identically equal to zero, 

 if the tei-ms are multiplied out. But if we 

 can assume the numerical values of the 

 ratios of the triangles to be known from 

 other sources and if we express in each of 

 the three equations the heliocentric rectan- 

 gular coordinates in terms of the geocentric 

 polar coordinates and of the solar co- 

 ordinates so that, for instance. 



[TiiTii 



['■I'-ii 



The coefficients of x here represent the pro- 

 jections of double the triangular areas 

 [n Tj] upon the yz plane. By dividing 



(p: COS ai COS Si — Xi) — (pn COS an COS 5n — Xn) 

 + F^~"j (piii COS am COS Si„ - Zni), 



then we arrive at three equations with the 

 geocentric distances as unknown quantities. 

 Now if the ratios of the triangles could be 

 known at the outset, it is evident that the 

 geocentric distances can be obtained by the 

 solution of these three equations. The 

 first approximation, depending upon the 

 degree of accuracy with which the ratios of 

 the triangles are introduced, is generally 

 referred to as the first hypothesis and the 

 accuracy of the geocentric distances and 

 therefore of the whole solution which de- 

 pends upon them is referred to as being of 

 the zero, first, or higher order with refer- 

 ence to the intervals or motions. The choice 

 of equal intervals always increases the 

 accuracy by one order. Simple as this 

 process seems in theory, it becomes very 

 complicated in practise, because in general 

 a first approximation can not be obtained 

 by merely using the ratios of the intervals 

 as numerical expressions for the ratios of 



