PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES, 141 
which will be denoted by A, and will be called the absolute longitude, or 
mean longitude for t=0. 
The six arbitrary constants which have now been defined are the 
elements which fix the absolute orbit of the disturbed body, namely, A 
(the absolute longitude), (the longitude of the absolute perihelion), © 
(the longitude of the absolute node), a (the protometer), « (the diastem- 
matic modulus), and 7 (the anastemmatic modulus). 
Having now described the form in which the solution is to be obtained, 
we can resume the consideration of the differential equations. 
First, we have to expand in a suitable way the disturbing function 0 
and the quantities P and Q. This is effected by means of infinite series, 
each term of which consists of a product of powers of the various small 
quantities such as 7, multiplied by a trigonometrical function of the 
longitude. 
Next, we have to substitute these expansions in the differential equations 
for p, s, and W, and integrate these equations. 
The equations for p and 8 are respectively of the forms 
dp ae era) 
7,3 t(1—B)ps= 3a, cos (ww By), 
dS as ? 
= b, cos (A,V—/3,); 
where the quantities A, are constants, but the quantities a,, b,, (,, contain 
the unknown variables. These equations are solved by processes of suc- 
cessive approximation ; only those terms are initially retained which have 
a considerable effect, and in this way the elementary part (p) is determined. 
A feature of Gyldén’s treatment of equations such as that given above for 
p is the use of the horistic function, which is a modification of the term 
containing the first power of the dependent variable, and is designed to 
ensure the convergence of the approximations. 
We may regard the arbitrary constants « and Tas arising from the 
integration of the equation in p, i and 0 as arising from that in z, a as 
arising from that in 8, and A as arising from that in W. 
The principal papers in which Gyldén’s theory has been developed will 
now be briefly noticed. In 1881 Gyldén published three short papers ! in 
French and German, and three long memoirs in Swedish.? The deriva- 
tion of the differential equations of the first Swedish memoir was simplified 
by Backlund 3 and Callandreau.! 
Further notes and criticisms on the early part of the theory of 
intermediate orbits were given in 1882 by Thiele® and Radau.® The 
1 «Sur la théorie du mouvement des corps célestes,’ @. R. xcii. pp. 1262-5; ‘ Sur 
lintégration d’une équation différentielle linéaire du deuxiéme ordre dont dépend 
l'évection,’ C. R. xciii. pp. 127-31 ; ‘ Ueber die Theorie der Bewegungen der Himmels- 
k6rper,’ Ast. Nach. c. p. 97. 
* ‘Undersékningar af theorien for himlakropparnas rorelser,’ Bihang till K. Sv. 
Vet.-ak. Handlingar, vi. and vii. I wish gratefuily to record my obligations to Mr. 
W. F. Sedgwick, late Scholar of Trinity College, Cambridge, who has kindly placed 
at my disposal a manuscript English translation of the Undersékningar, with many 
corrections of his own. 
8 « Ableitung von Professor Gyldén’s Differentialgleichungen fiir die intermediire 
Bewegung,’ Asi. Nach. ci. pp. 19-22; Professor Gyldén’s ‘Neue Untersuchungen iiber 
die Theorie der Bewegung der Himmelsk6rper,’ Copernicus, ii. 
‘ «Sur la théorie du mouvement des corps célestes,’ C. #. xciii. pp. 779-81. 
° ‘Ueber Professor Gyldén’s intermediiire Bahnen,’ Ast. Nach. cii. pp. 65-70. 
® «Sur un pojnt de la théorie des perturbations,’ C. RB. xcy. pp. 117-20. 
