72 A. Mukhopadhyay — JDifferential Equations of Trajectories. [No. 1, 



at the base of tlie scutellam ; one similar spot, oblong and transverse 

 on the disc of each hemelytrum : body beneath yellow with a black 

 bronzed spot on each side of the mesostethinm ; a narrow band of the 

 same colour at the base of the venter, and a row of five similar spots on 

 each side : the abdominal point reaches only the insertion of the inter- 

 mediate feet {G. incarnatus, Am. & Serv.). Long, 25 — 30 mill. 



Yar. h. : — Large ; head with antennae deep black ; pronotum orange, 

 with the anterior margin deep black : scntellum orange, immaculate : 

 hemelytra orange with a median fuscous spot : wings fuscous : margin 

 of abdomen variegated with orange and black : feet deep black ((7. 

 aiirantius, Fabr.). Long, 25-30 mill. 



Var. c. : — Scutellum, hemelytra and pectus immaculate. Ceylon. 



Reported from Corea, Japan, Java, Sumatra, Borneo, Siam, Malac- 

 ca, Singapore, Tenasserim, Ceylon, Madras, Bombay, Bengal, Pondicher- 

 ry, Silhat, Assam. The Indian Museum has specimens from Tenasserim, 

 Assam, Sikkim, Calcutta, Karachi, Malabar. Varies in colour from a 

 sordid yellow, to orange and a bright maroon red, with and without the 

 black spots. 



II. — A General Theorem on the Differential Eqitations of Trajectories. 

 — By AsuTOSH MuKHOPADHTAT, M. A., F. R. A. S., F. R. S. E. 



[Received November 17tli ; — Bead December 7tli, 1887.] 

 Contents. 

 §. 1. Introduction. 



§. 2. Statement and demonstration of the theorem. 

 §. 3. Application of the theorem to Mainardi's problem. 

 §. 4. Other applications of the theorem. 

 §. 5. Some applications of Conjugate Functions. 



§. 1. Introduction. 

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

 was read at the last May meeting of the Socieiy, {Journal, 1887, Vol. 

 LVI, Part. II, pp. 117—120; Proceedings, 1887, p. 151), I pointed out 

 that Mainardi's complicated solution (reproduced by Boole) of the pro- 

 blem of determining the oblique trajectory of a system of confocal 

 ellipses, was equivalent to a pair of remarkably simple equations which 

 admitted of an interesting geometrical interpretation. Believing, as I 

 firmly did, that every simple mathematical result could be established 

 by a correspondingly simple process, I naturally thought it worth while 

 to re-examine the whole question, to see if the very artificial process of 



