422 



SCIENCE. 



[X. S. Vol. XXI. Xo. 5.33. 



not for both solutions. So far is he from 

 using it as a criterion. 



It Avould, indeed, appear that very sim- 

 ple reasoning indicates [vv] as the right 

 criterion. If we consider observation 

 equations of the general form: 



Hx, y, 2, n,) = 0, 

 the ordinary solution determines x, y, z, 

 so as to make [w] a minimum. If there 

 exists a doubt as to whether the form of 

 the function </> should be either or 

 this fact simply transfers ^ to the list 

 of unknowns, and we must so determine 

 X, y, ■■ ■, as to make [vv] a minimum. 

 We shall do this if we make two ordinary 

 least squares solutions for and <f)2, the 

 only possible values of <p, and prefer that 

 solution which gives the smaller [vv]. 

 Since the other criterion may give an op- 

 posite result, that other criterion must be 

 wrong. 



It may be of interest to add to the above 

 a remark concerning the attractive result 

 obtained by Mr. Midzuhara in his equa- 

 tion (13).* This result is: 



(13) 



where iv is the value of the new unknown, 

 obtained in the solution of our equations 

 (2) and its weight from the same solu- 

 tion. Mr. Midzuhara gives a somewhat 

 extended demonstration of this equation 

 (13) ; it may, however, be obtained almost 

 directly from a principle demonstrated by 

 Gauss in 'Elementis Ellipticis Palladis.'f 

 It is there shown that if jit be the number 

 of unknowns, and if the normal equations 

 are solved by the Gaussian method of 

 elimination : 



[w] = [nil • //J, 



where [■nn-/x] denotes the usual Gaussian 

 auxiliary. In the present case, if there 

 are /x unknowns in equations (1), and 



* Astr. Jour., 568. 



V VVerke,' Vol. 6, p. 22. 



H -\- 1 in equations (2), we shall have, at 

 the end of our Gaussian elimination : 



[pp-f^lw 4 [pn- m] 0, 

 [ran • fi] 



[WW • (/i i- 1)]. 



But, according to Gauss's principle: 



[nn-fi] = [t'D]„ 



[nil • -I- 1)] = [ir]^, 



and, as usual: 



[nn • (// -I- 1)] = [nw •//]- 



[pp /A 



Therefore 

 But: 

 and : 



[PPF] 



so that: 



[ PP ■ f'l — -Pw — weight of 

 [n.],-[iT],^,c'P,,. 



This is Mr. Midzuhara 's equation (13). 



Tables for the Eeduction of Astronomical 

 Photographs : Harold Ja.coby. 

 In 1895 the writer published a paper en- 

 titled 'On the Reduction of Stellar Photo- 

 graphs, with Special Reference to the 

 Astro-Photographic Catalogue Plates.'* As 

 indicated in the title, the method there de- 

 scribed was intended primarily for the re- 

 duction of large series of plates made at 

 the same declination. But ordinary stellar 

 photographs intended for star-cluster cata- 

 logues, solar or stellar parallax, etc.. usu- 

 ally involve so few plates of a single 

 declination that it is not economical to pre- 

 pare the kind of special tables suitable for 

 a photographic catalogue of the whole 

 heavens. IMoreover, Contribution 10 of the 

 Columbia Observatory has long been out 

 of print, so that it is now impossible to 

 supply copies to those asking for them. 



*'Contril). from the Obs. of Cohiiiihia Coll." X'o. 

 10; and in French, Bull. Com. Perm., Tome III. 



