1888.] A. Mukhopadhyay — Differential Equations of Trajectories. 09 



is irrelevant. We conclude, therefore, that, of the two solutions to 

 which Mainardi's result is really equivalent, only one is relevant ; the 

 other being wholly extraneous, as belonging to the oblique trajectory of 

 the orthogonal system of confocal hyperbolas ;* and, it is easy to dis- 

 criminate which of the two solutions given by the quadratic 



leads to the relevant solution ; for we have seen that the solution in 

 point is furnished by 



A 



now it is evident geometrically that 



which shews at once that 



hfx , h\ 



A IX 

 it follows, therefore, that the smaller of the two roots of the quadratic 

 in M is the proper value. We come to the conclusion, therefore, that in 

 Mainardi's system 



i_ All 



-2ni^n-\/~-l-\-log ^^_ ^ = 0, 



V ^M / ivr 



the smaller root of the quadratic in M gives the oblique trajectory of the 

 system of confocal ellipses, while the greater root furnishes the oblique 

 trajectory of the system of confocal hyperbolas. I am not aware that 

 the real character of the two solutions to which Mainardi's result is 

 equivalent has been before distinguished as above. 



Lastly, it is sufficiently obvious that the values of X, fx given by 

 either of the above systems may be geometrically represented by a 

 construction closely analogous to what is given in my former paper 

 mentioned at the beginning of this memoir. 



lOth December 1887. 



* Instances of a single solution resolving itself into two, are by no means rare ; 

 for example, in the case of the conic 



aar^ + 2hxy + hy^ + 2gx + 2fy + c = 0, 

 this equation is really equivalent to the two 



&i/ = - {hx +/) + \/(h^-ab}x^ + 2{hf~hg)x-i-{p-0c) 

 ly = - {hx+f)^^{h^ - ab)x^ + 2{hf- bg)x+{f^ - be) 

 Bnt the present case is distinguishable from the case of the conic, inasmuch as we 

 have here one of the solutions irrelevant, while, in the case of the conic, both the 

 solutions are relevant, the compound solution being reproduced by multiplying 

 together the resolved solutions. 



