c-T 



its Analogues, and the Calculus of Imaginaries. 275 



relation is equivalent to 



(m 2 + im^j (m 2 — im v ) = 0, 

 where i is such an operation that 



Thus m 2 has either of values + im v and upon the introduction 

 of these, the value of U becomes 



or 



XAe m ^(cos m^ + * . sin rn-^y) 



i + . K ." . ' - («i) 



L SBe^-^cos m^y—i . sin m^) J 



which is a general solution of (I.) in the first form. It is evi- 

 dent that this solution may be condensed, but we shall retain it 

 in its present shape for tha sake of symmetry. 



Now since the arbitrary constants A, B, and m x are indepen- 

 dent of each other, we may obviously substitute for A and B 

 arbitrary functions of m v and the solution just found may then 

 be put under the shape 



P/©(«ll)«W; cos m x y . dm{\ 



L/^ r (w ] )e m i* . sin m } y . dm l J 



the limits of the integrals being supposed independent of x and y. 

 At first sight this might be regarded as a solution in the second 

 form, that namely of a definite integral. It seems, however, 

 more just to consider it as merely another shape of the first form, 

 since the limits are indeterminate. 



The general solution in the third form is given by 



u=(— -*•— v 1 (* +% d y i 



\dy dx) ' \dy dx) 

 or is 



U = (j>(x + iy)+^(x-iy), (c t ) 



and the coincidence of the previous with this is evident. 



2. Similarly, a general solution of the higher equation, ren- 

 dered famous by its connexion with the name of Laplace, 



d*V d*V d*V A , TTN 



^ + ^ + ^=° < IL > 



is given by 



where m v m 2 , m 3 are connected by the corresponding relation 

 m x 2 + m£ + m 3 2 = 0. 

 T2 



