OP ARTS AND SCIENCES. 15 



fill whether the observations yet made are sufficiently exact to prove 

 a connection between the components. To establish this proposition, 

 and also as an example of a convenient means of distinguishing a 

 binary star from one which is optically double, the following investiga- 

 tion is given of the more important observations of 61 Gygm. We 

 cannot conclude that a star is binary unless the path described by one 

 of its components appears to be concave with respect to the other. 

 If the motion appears to be rectilinear, it is more probably that due to 

 the proper motion of one of them, or rather to the combined effect of 

 the proper motions of both. On the other hand, if the path is con- 

 vex, it is extremely probable that there is a real connection between 

 the two, as there is no instance known of a star describing a curved 

 path due to proper motion alone. The motion, if rectilinear, should 

 also be uniform, while, if curved, the motion should be most rapid 

 when nearest the other star. The law that the area described by the 

 radius vector is proportional to the time, cannot be used to distinguish 

 between those motions, since it will apply to both. 



Suppose that the measures are transformed to a system of rectan- 

 gular co-ordinates, having one component as the origin, and the axis 

 of X nearly parallel to the path of the other component. Except 

 for the accidental errors, the value of y, if the motion is rectilinear, 

 should be the same for all the observations from the beginning to the 

 end of the series. If the axis of X. is not exactly parallel to the line 

 of motion the values of y should increase slowly from one end of the 

 series to the other. If they are corrected by an amount which will 

 be proportional to the time, this variation should disappear. If the 

 star is binary, however, the value of y will vary, in general having 

 its greatest value during the middle of the period, and being smaller at 

 the beginning and end. 



The values of a:, if the motion is rectilinear, will vary uniformly 

 with the time, and, if corrected by a constant, plus another constant 

 multiplied by the time, will leave residuals that are very small. If 

 the motion is curved, on the other hand, this condition will not be 

 fulfilled. 



A reduction of the observations of 61 Cygni is given in Table VI. 

 Of the measurements made during the last half-century only those 

 made by the Struves and by Dembowski have been employed. The 

 position angles are first corrected for precession and reduced to the 

 epoch of 1880 by the formula 



0°.00557 sin « sec b (t — 1880) = — 0°.005 {t — 1880). 



