338 



SCIENCE. 



[N. S. Vol. III. No. 62. 



in most cases of which the line of apsides is 

 vai'iable, and considers the situation and 

 density of the points of intersection of these 

 curves with themselves. 



The idea of an absolute orbit of a plane- 

 tary body is this: an oval symmetrical with 

 regard to an axis movable in space. While 

 the axis remains constant in length (the 

 half of it is called theprotrometre), the ve- 

 locity of its motion may vary, and the di- 

 astem may also vary. Prof. Gj'lden, how- 

 ever, admits into the expressions of these 

 variations only terms whose period would 

 become infinite did the planetary masses 

 vanish. These terms he calls elementary. 

 But elementary terms in the diastem and 

 the longitude of the perihelion can produce 

 terms in the coordinates having periods 

 which differ but little from the time of 

 revolution of the planet. These are also 

 called elementary terms. But the two 

 classes are distinguished, the first as being 

 of the type (A), and the second as of the 

 type (B) . In all the formulas relative to 

 this matter the author insists on keeping 

 the arc described by the radius as the inde- 

 pendent variable. 



The coordinates are only approximately 

 given by the preceding apparatus of expres- 

 sions. They must then have certain com- 

 plements added to them ; these, however, 

 are all composed of terms which would van- 

 ish with the planetary masses. 



In deriving the elementary terms in the 

 radius of a planet through the integration 

 of a linear differential equation of the sec- 

 ond order, Prof. Gyld6n attaches much price 

 to his method of establishing the conver- 

 gence of the series formed by the successive 

 terms. As the latter are obtained through 

 division by divisors of the order of the 

 planetary masses, it might be feared that 

 some of them would turn out to be very 

 large. But the author prevents this by re- 

 taining in the coeificient of the dependent 

 variable in the differential equation a 



quantity equivalent to the sum of th« 

 squares of all the coeflicients in the inte- 

 gral. This is named the horistic or limiting 

 function. It is plain such an expression 

 could be introduced in the mentioned co- 

 efficient, provided that the linear equation is 

 the truncated form of an equation contain- 

 ing the cube of the variable. And in the 

 problem of planetary motion the approxi- 

 mations may always be so ordered that this 

 shall be the case. 



With regard to the coordinate which 

 exhibits the departure of the planet from a 

 fixed plane, Prof. Gylden does not greatly 

 deviate from the procedure of Hansen in 

 following the displacement of the instan- 

 taneous plane of the orbit. Only here, as 

 in the preceding treatment of the radius, 

 he would sharply distinguish the elemen- 

 tary and non-elementary terms. 



At this point is introduced certain new 

 nomenclature. As before we had diastem, 

 now we haveanastem to denote the product 

 of the radius and the sine of the inclination; 

 and what has generally been called the true 

 argument of the latitude is here called the 

 anastematic argument. Any angular mag- 

 nitudes which are constantly moving 

 through the circumference are astronomic 

 arguments; and when they have the same 

 mean velocity of rotation they are isoki- 

 netic ; and isokinetic arguments are hom- 

 orhythmic when, in each revolution through 

 the circumference, they always retake to- 

 gether the same corresponding points. In 

 like manner, the true anomaly is the dias- 

 tematic argument, and we have diastematic 

 and anastematic coefficients and moduli. 

 It will be seen from this that Prof. Gylden 

 does not shrink from imposing on us the 

 labor of leai'ning new terms. 



Thus far we have been engaged in de- 

 riving the equations of the path followed by 

 a heavenlj' body ; it remains to show how 

 we may find tlie point on that path occupied 

 by the body at a given moment. There is 



