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IX. On the Integration of Partial Differential Equations of 

 the Third and Higher Orders. By Robert Moon, M.A., 

 Honorary Fellow of Queens' College, Cambridge* '. 



THE following method of deriving the first integrals, where 

 such integrals exist, of the equation 



R^+ S ^^+ T ^T-2+U^3==Y . . (1) 



dx 6 dx 2 dy dxdy 2 dy z v y 



(R, S, T, U, Y being functions of x and y only), and of deve- 

 loping the conditions under which alone the equation will 

 admit of such integrals, is capable of being extended to Partial 

 Differential Equations of the higher orders involving two 

 independent variables. 



Assume (1) to be satisfied by 



f(xyrst) = 0, (2) 



where r, s, t respectively have their usual values. 

 Differentiating (2) with respect to x and y, we get 



j!f __ /'(*) d * z /'(*) d * z ../(*) 

 *'' dx d f{r)dx 2 dy f{r)dxdy 2 f'(r)' 



tl = _/V) ** f(s) dh f(y) 

 dtf fit) dx 2 dy f{t)dxdy 2 f'(t) ] 



the substitution of which in (1) gives 



o n fM _*l ft/'?*) ** v f'(<°) 



f'(r)dx->dy *f(r)dmd? K f'(r)' 



_ f(r) J?z_ f(s) _^_ _ n /^) 



U /'(«) d^ dy U /'( t) dx dtf U f(t) ' 



_±_ Q Z i T z ~\r 



dx 2 dy dx dy 2 ' 



d 3 z 

 in order to which we must have the coefficients of ■ , , T . 



dor dy^ 



* Communicated by the Author. 



