( 467 ) 



d^ z— dM^ -\- ,v sin nt^ — y cos nt^ ...... (a) 



do da d^ , «^ , ^ ,,, 



~ = \ cos^ nt, i" A sin 2nt. . dM. — 



f «' \ f . «' . . A 



— ^ A' j cos nt^ sin 2nt^ sin nt^ 1 — ^y\ sin nt^ -\ sin 2 nt^ sin nt^ | . (b) 



As these differential expressions have led several astronomers ^) 

 into eri'or, we w^ill derive them in still another way. 

 From : 



^ =z V -\- (a 

 we get : 



d^ ■=: dv -{- doi. 



In the circular orbit v = M ; in the elliptic orbit this becomes : 



u = 1/ -f 2e sm 1/ + . . . -{- dM^. 



If we substitute M^ nt^ — oi, and put c/w =^ 0, \\q get, neglecting 

 higher powers of e : 



d^ = dÈl^ -\- X sin nt^ — y cos nt^ 

 If in : ?^ = *'' 5*^'*^ /? -|- r" cos'^ i cos^ /?, 



then, neglecting higher powers of i' : 



do dv T^ ?'" 



— — — -X- — -sin2^dS -^ cos ^^. i" . . . . (c) 



Q r 2q^ 2q^ 



In the elliptic orbit we have : 



ail—e') 



= a^ 4" da — ae cos {3 — co) -!-••• = 



1 -{- e cos V 



1 1 



■= a„ -\- da a x cos nt, ay sin nt, . . . 



Therefore : 



1 1 



dr z= da ax cos nt, ay sin nt, 



2 ^2 



and, substituting this in (c), we get the expression already given of 

 do 



1) Dr. Myers puts d/3 = for ^^ = and at the same time dM^ = ; this is 

 incompatible with (a). Prof. Hartwig, in his paper: "Der veranderliche Stern vom 

 Algoltypus Z Herculis" (Bamberg 1900) p. 39, puts sin {P- ^) cos {P- Sh) sin i tg i 



= for i = 90°, whereas, accordmg to our formulas, it becomes — ^r— - sin 2S. 



2^^ 



(See also A.N. 3644). 



Dr. Pannekoek quotes another instance in his Thesis on Algol (p. 22 — 3). 



32 



Proceedings Royal Acad. Amsterdam. Vol. X. 



