ON TERNARY DIFFERENTIAL EQUATIONS. 10] 



XV. On Ternary Differential Equations. 

 By Robert Rawson, Esq., Hon. Member of the Society. 



Read February 6th, 1877. 



I. The following observations have been suggested by 

 reading the interesting communication to the ordinary 

 meeting of November 28th, 1876^ '^On Ternary Differen- 

 tial Equations/' by Sir James Cockle. 



With the view of comprehending fully the nature and 

 importance of the step advanced by Sir James Cockle in 

 the communication above referred to, it will be necessary 

 to narrate briefly the exact state of our present knowledge 

 of this subject, as given by Boole and other eminent 

 writers on differential equations. 



The substance of this knowledge may be given in two 

 propositions, as follows : — 



First. That the differential equation 



Vdx + Q,dy + ^dz = o (i) 



(where P, Q, R are given functions oix, y, z) has a single 

 solution (or, in other words, there is a single relation be- 

 tween the variables x, y, z which will satisfy it), provided 

 the following conditional equation obtains : — 



\dz dy J \dx dz / \dy dx / 

 n is called by Sir James Cockle the discriminoid. 



Second. If the conditional equation (2) does not obtain 

 by virtue of the given functions P, Q, R, then the equation 

 (i) has a dual solution — or, in other words, there are two 

 relations between the variables x, y, z which vvill satisfy it. 



