vJJSQ A. Mukhopadhyay — On Trajectories. [DeC. 



of Laos: also an old Tibetan Sanskrit Dictionary, brought from Lhasa, 

 arranged in alphabetical order, and written in the U-ine, or headless 

 character of Tibet. 



The Rev. Fb. Lafont made the following remarks on the musical 

 instrument exhibited by Babu. Sarat Chandra Das : — 



The Phong is a free-reed instrument, of very sweet tone and very 

 ly made. The fourteen pipes are most carefully tuned to a full 

 natural octave in the middle key and re, mi, fa of a treble, with the la 

 and di of a lower tone. The perfect chord do, mi, sol, do, is particularly 

 good, and the two pipes tuned in sol are in perfect unison. It is inter- 

 esting to see the careful manner in which the length of each pipe is 

 adjusted by deep longitudinal slits, cleverly corrected for pitch by tiny 

 little bits firmly cemented. The performer has only to blow gently 

 through one universal mouth-piece carrying the wind to all the pipes, 

 but allowing only those to speak where the little hole made above the 

 reed is covered by the fingers of the musician. It is difficult to believe 

 that the instrument is the result of pure native Siamese ingenuity, I 

 feel inclined to think that a European musician had a hand in it. 



The following papers were read — 



1. A general Theorem on the Differential Equations of Trajectories.—^ 

 By Babi5 Asutosh Mukhopadhyay, M. A., F. R. A. S., F. R. S. E. 



(Abstract.) 



In a paper on " The Differential Equation of a Trajectory," which 

 has been published in Part II of the Journal for 1887, and an abstract 

 of which has already been given in the present volume,* the author 

 pointed out that Mainardi's complicated solution of the problem of 

 determining the oblique trajectory of a system of confocal ellipses, 

 is equivalent to a pair of remarkably simple equations, which admit of 

 an interesting geometrical interpretation. On re-examining the whole 

 question to see if the very artificial process of Mainardi, by no means 

 less complicated than his result, could be materially simplified, the 

 author has been led to a very general theorem on the differential equa- 

 tions of trajectories, which is established and illustrated in the present 

 paper. The paper is divided into five sections, of which the first is 

 introductory. The second section contains the enunciation and demon- 

 stration of the theorem, the chief characteristic of which is the property 

 that whenever the coordinates of a point on any curve can be expressed 



* The equations on page 151 are wrongly printed ; they should have been 

 x = h cos <p. cosh n ( A + <p) 

 y = h sin (p. sinh n ( A + (f>) 

 where /t 8 = a 2 — I 2 , so that h is half the distance botween the foci. 



