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SCIENCE 



[N. S. Vol. XLV. No. 1171 



the triangles. It is necessary to introduce 

 at the outset one or more of the terms in- 

 volving the inverse powers of the helio- 

 centric distance r and its derivatives, and 

 these can not be known until the geocentric 

 distances have been obtained, r being de- 

 rived from the triangle which has at its 

 vertices the observer, the sun, and the body. 

 The angle at the observer is known by ob- 

 servation, the distance of the sun from the 

 earth is known, and p being assumed, r 

 may be found. But since neither r nor P 

 is known at the outset, the solution must 

 be accomplished by trial and error. Here, 

 as before, the method of differentially cor- 

 recting the first approximation on the basis 

 of the difference between initial and final 

 values in a trial is very effective. Thus 

 the first hypothesis involves a series of 

 trials for the solution of r and P, and it is 

 accurate to zero, first, or second order, and 

 so forth, according to the number of terms 

 in the ratios of the triangles introduced in 

 the first set of trials for the distances, 

 which trials become, of course, the more 

 complicated, the more terms are introduced. 

 A practical limit is thus set at once. The 

 second hjT)othesis depends upon the com- 

 putation of the remaining terms of the 

 series in the ratios of the triangles on the 

 basis of values derived from the first 

 hypothesis. "While these values may be 

 sufficient for the higher terms the lower 

 terms taken into account in the first hypoth- 

 esis still remain inaccurate since they do 

 not contain the final numerical values of 

 the unknowns. This is important because 

 it involves successive resubstitution of the 

 improved values in all terms of the series. 

 We shall see later that these complicated 

 manipulations have recently been removed 

 by Charlier by completing a purely analyt- 

 ical solution proposed by Lagrange. In 

 passing from one hypothesis to the next it 

 is necessary, as previously stated, to calcu- 



late the remaining terms of the series repre- 

 senting the ratios of triangles. Oppolzer 

 ingeniously computes the whole remainder 

 in a closed form, but in determining the 

 numerical value of the closed remainder 

 must perform successive approximations, 

 as I say, merely to get the remainder. 

 These approximations involve the compli- 

 cated expression of the ratio of a sector of 

 a conic to the corresponding triangle. 

 Thus we see that in Oppolzer 's formula- 

 tion, which is the most accurate yet pro- 

 posed, it is necessary first: to undertake 

 several hypotheses; secondly: within each 

 hypothesis to perform a number of trials 

 for the distances; and, thirdly: in passing 

 from one hypothesis to the next to perform 

 approximations involving the ratio of sector 

 to triangle. The application of the method 

 of differential correction, so successfully 

 applied in the trials for the distances, in 

 place of these several cycles will take up 

 in one operation all of these cycles of ap- 

 proximation as will be referred to later. 



It is not necessary to go into the various 

 and numerous devices which have been pro- 

 posed during the past century to facilitate 

 the various cycles of approximation re- 

 ferred to. Be it sufficient to say that the 

 highest degree of accuracy has been ob- 

 tained in this country by Gibbs in his vector 

 method, which in the first hypothesis takes 

 account of terms of the fourth order in the 

 ratios of the triangles. But although this 

 method is the most accurate of all the so- 

 called methods in the first hypothesis it un- 

 fortunately requires too large an amount of 

 numerical work in the first approximation 

 and does not readily lend itself for applica- 

 tion to a second hypothesis. It has, there- 

 fore, failed to come into universal use. 



I have already referred to the methods 

 hitherto described as "the older methods." 

 They have been set down in various excel- 

 lent formulations, particularly by Klinker- 



