154 



SCIENCE 



[N. S. Vol. XXXI. No. 787 



Reflex Zenith Tube of the Flower Observatory," 

 by C. L. Doolittle. 



21. " Visual Observations of Variable Stars at 

 the Harvard College Observatory," by Leon 

 Campbell. 



Professor Keyser's vice-presidential address ap- 

 peared in full in the December 31 number of 

 Science. In the absence of their respective au- 

 thors the paper by Professor Todd and those of 

 Messrs. Wetherill and Slipher were read by title, 

 while that of Mr. Stewart was presented by Dr. O. 

 .J. Klotz, Ottawa, Canada. The abstracts which 

 follow bear the same numbers as the correspond- 

 ing titles in the preceding list. 



2. The problem of finding your geographical 

 position in a balloon from observations of the sun 

 is very different from the same problem on board 

 ship, for this reason, that in a balloon there is no 

 dead reckoning. The method used on board ship 

 of observing two altitudes of the sun at two dif- 

 ferent hours of the day can not be applied, for 

 the two Sumner lines have to be shifted so as to 

 correspond to the same moment and this can only 

 be done by dead reckoning. In a balloon, there- 

 fore, the only way of getting your geographical 

 position from the sun is by observing both alti- 

 tude and azimuth at the same time. Now the 

 accuracy with which the azimuth of the sun may 

 be observed is rather small; it would be diflScult 

 to obtain it within less than one tenth of a degree. 

 Therefore the reduction of the observations need 

 not be very accurate, either. At the same time it 

 is essential that the reduction should be made 

 very quickly. For the time since the moment the 

 observations were taken introduces an uncertainty 

 that may be expressed by the area of a circle 

 whose radius is equal to the distance through 

 which the balloon may have traveled. One nat- 

 urally would therefore turn to graphical methods 

 for the reduction of the observations. The reduc- 

 tion consists in finding the latitude <t> and the 

 hour angle t from the declination 5, the azimuth 

 a and the altitude h. Professor Runge proposes 

 to find first the latitude </> from 8, a, h and then 

 the hour angle t, from 5, o, (p. In both cases we 

 have to deal with the representation of an equa- 

 tion between four variables. Both of these equa- 

 tions may be written in the following form: 



f{p) +h(r, s)g{q) =k{r, s) 



where p, q, r, s denote the four variables. That 

 is to say, two of the variables enter the equations 

 in functions of their own f{p), g(q) and the 

 equation is linear in these functions, the coefii- 



cients being any functions of the other two vari- 

 ables. Equations of this kind may be represented 

 graphically by the " mgthode des points aligngs " 

 of Maurice d'Ocagne* taking f(p) and g{q) as 

 line coordinates. I propose making f{p) equal 

 to the ordinate of the point of intersection of the 

 straight line with the axis of ordinates and 

 g(q) equal to the gradient of the straight line, 

 that is, the tangent of its angle with the axis of 

 abscissa. In that way the rectangular coordinates 

 ot the point whose equation in line coordinates 

 is the given equation, become: 



x^h(r, s) and y^k{r, s) . 



For any given value of p, the different values of 

 q correspond to straight lines that form a pencil 

 of rays, whose center is on the axis of ordinates 

 at the particular value defined by p, and any 

 alteration of p would simply shift the center 

 along the axis without altering the pencil of 

 rays in any other way. The whole diagram may 

 therefore be obtained by drawing two figures, one 

 containing the curves r =: const and s = const, 

 the other containing the pencil of rays, and 

 placing these two figures in the proper way, one 

 over the other. It so happens in our cases that 

 the variable p is the declination of the sun, which 

 during the ascent of a balloon may be regarded as 

 constant. The aeronaut would therefore merely 

 use a definite superposition of the figures. They 

 are photographed on transparent plates and a 

 blue print is taken by copying the plates one after 

 another on the same paper in the proper position. 

 The aeronaut has one blue print to read off the 

 latitude and a second one to read off the hour 

 angle after he has found the latitude. The equa- 

 tiong are: 



( 1 ) sin 5 + cos cos h cos a = sin </> sin h, 



(2) tan S + sec <p sin t cot a = tan </> cos *. 



In the first equation the curves = const and 

 h = const are the ellipses 



X = cos cos h, y^ sin <p sin Ti. 



In the second equation the curves 0^ const and 

 t = const are the oonfocal ellipses and hyperbolas 



01 = sec sin t, y = ta.n tp cos t. 



3. Newcomb has shown that there is a differ- 

 ence between the observed and the theoretical 

 positions of the moon which can be roughly repre- 

 sented by a term of period about 270 years and 

 coefficient 13". In this paper Professor Brown 



' Maurice d'Ocagne, " Traits de Nomagraphie." 



