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XIII. On the Integration of the General Linear Partial Differen- 

 tial Equation of the Second Order . By R. Moon, M.A.j Ho- 

 norary Fellow of Queen's College, Cambridge *. 



I PROPOSE to show in the following paper that when the 

 equation 



°= R £ +s £ y +T S +v - • • • « 



is derivable from a single partial differential equation of the first 

 order, such single equation will be of one or other of the three 

 following forms, viz. : — 



I. q=f(ocyzp) i 



where /satisfies both the equations 



0=f'(p) -m u 



Q=mJ\x)-f[y) + (m,p-/)f(z) -(J) _ 



where m v m 2 are the roots of the equation 



•0 = T q=f . m? + S ?=/ . m + R, =/ . 



II. q=f{xyz) } 

 where we have simultaneously 

 = R ?=/ , 

 = T q=f .f'(y) + S q=f .f'(x) + (T q ^.f^ q=f . P )f(z) +V . 



III. p=f{ocyz), 

 where we have simultaneously 



0=R P=/ ./'(*) + S, =/ ./'%) + &„=/. f +&,=,. q)f(z)+Y p=f . 



I. Let definite values be assigned to any arbitrary functions 

 which may be contained in the single equation of the first order 

 from which (1) is derivable, so that it may be represented by 



q=f{xyz P ), (2) 



unless indeed it do not contain q } a case hereafter to be con- 

 sidered. 



Differentiating (2), we get 



0=|f-/'(^)^-/„ .... (3) 

 * Communicated by the Author. 



