Symmetrical Integration. 351 



from it satisfy the given equation, and the characters of these 

 derivees, we see at once that, similarly, the solution of the partial 

 differential equation in three independent variables, 



d 2 u d% d*u _, 

 dx dy dz dx dy dz~~ ' 

 is 



u = k*(x + a) (y + <fm) (z + %a) i 



% being a new arbitrary function. 



Similarly, we see that the solution of the partial differential 

 equation in four independent variables, 



d s u d s u d 3 u d s u , 



dxdydz dwdxdy dzdwdx dydwdz ' 

 is 



«= k*(x + «)(?/ + <f>ot) (z + %a) (w + i/ra), 



where i|r is a third new arbitrary function. 



6. Let it be required to determine the values of u and v, being 

 given the system of simultaneous partial differential equations, 



aj) x u + bfiyU + cj) z u + a^) x v + b^DyV + c^) z v = k x u + & 2 v,l 

 fljl) f « + bfiyV + CjD^v -f ^D^w + b^DyU + c 2 D,~w = &, v + >fc 2 w, J 



where the coefficients a ly b v c v &c. are supposed to be constants. 

 These equations may be thrown into the form 



(aj), + byDy 4 CiD*— k Y )u + (<? 2 1)^+ 6 2 D y + c^— & 2 ) 0=0/1 



% (a 1 T>s + bJ) & + c 1 I) z —k 1 )v+(aJ) x + bfiy + CiP as --k^u = O i J 



whence we derive 



fpfi 9 + bfi y + pj^| - k x f . u - (« 2 D, + bj) y f c 2 V z - & 2 ) 2 . u = 0. 



This equation may evidently, in general, be reduced to the form 



(« 1 D # + /3 1 D y + 7l D,-l)(« 2 D, + ftD y + 7 9 D 1B -l).tt=0, 



where a x , &, y^ &c. are known constants. And the solution 

 required is 



* JL £. ?L JL ?L 



^=^(6% e\ <?^)-f-\P\(e a ve&, e72 )> 



where $> v ^ are arbitrary homogeneous functions of the first 

 degree. 



The value of v is then to be had by substitution of the ex- 

 pression for u just found, in either of the given equations. 



Trinity College, Dublin, 

 October 1860. 



