214 



SCIENCE 



[N. S. Vol. XL VI. No. 1183 



and a bird must move forward to give the 

 same apparent displacement of objects against 

 the horizon. It is the purpose of the follow- 

 ing note to derive an analytic expression for 

 this curve. 



Consider first the ease of lateral vision. Let 

 A be the starting point of the bird, and let the 

 two objects, Ai and A2 in the original axis of 

 vision be at the distances a^ and a,, respectively, 

 from A. Let y be the distance that the bird 

 moves forward, and a the angle that is sub- 

 tended at its eye by the distance AiA2. (See 

 Fig. 1.) Then 



(5) 



sin (a -f 7 -f 5) 



(1) tan(a-h^)=^, tan0 = -^, 



where ^ is defined in the figure. Using the 

 trigonometric formula for the tangent of the 

 sum of two angles, and replacing tan P by its 

 value from the second equation of (1), we get 



y tan a + ai _ 02 

 y — Ui tan a y ' 



(2) 



Solving this for y gives 



(3) 2y tan a — 02 — ai ± VCaj — Qi)- — 4aia^ tan^ a. 



In taking up the ease of frontal vision, it is 

 necessary, as Mr. Trowbridge states, to have 

 a deflection between the line connecting the 

 observed objects and the direction of the man's 

 motion. Designating the angle of deflection 

 by S, and the distance that the man moves from 

 Ahy X (see Fig. 2), we have by the law of sines 

 sin (7 +,i) 



(4) 



sin 7 



cos S + 00% 7 sin 5, 



where again AAi = aj, AA! = a:, and a is the 

 angle subtended at the eye of the observer by 

 AAi. The angle 7 is defined in the figure. 

 Also 



02 sin (a + 7) 



By using the value of cot 7 obtained from (4), 

 we can easily eliminate 7 and reduce (5) to 



,„, X _ X sin {a + d) — oi sin a 



Oa a sin q; — ai sin (a — 6)' 



Solving for x gives 



(7) 2x tan a = a + Va^ — iaiOi tan^ a, 



where 



a = {02 + ai) cos 5 tan a + (a^ — ai) sin 5. 



Equations (3) and (Y) then are parametric 

 equations of the equal parallax curve. 



In plotting the curve of the practical prob- 

 lem we assign the values x = 0, y = for 

 a == 0. To a value of a slightly greater than 

 zero will correspond two values of x from (7) 

 and two values of y from (3). It is easily seen 

 that for the practical problem the smaller of 

 these must be chosen in each case; that is, we 

 must use the negative sign before the radicals 

 in (3) and (7). For Mr. Trowbridge's curve 

 the special values Oi = 1,000, 02 = 2,000 must 

 be assigned, and in all instances 5 must of 

 course be known. Paul R. EroER 



Washington Univeesitt, 

 St. Louis, Mo. 



A PREDECESSOR OF PRIESTLEY 



To THE Editor of Science: The notice of 

 the Priestley Memorial in the issue of Science 

 for August 17, 1917, reminds me of the best 

 chemical joke I have ever heard. I can hardly 

 forgive the " new chemistry " for having 

 spoiled it. At our Brown University club 

 dinners in Philadelphia we never have any 

 wine. Many years ago when water was " HO " 

 the late Eev. Dr. H. Lincoln Wayland, the 

 best wit I ever have known, after a very happy 

 eulogy of water, ended his after-dinner speech 



