274 Mr. R. Carmichael on Laplace's Equation, 



tions, it is objectionable from the difficulty of determining the 

 arbitrary functions. It may be observed that this last form is 

 unique, whereas in each of the two preceding cases we have 

 obviously as many general solutions as we can obtain particular 

 solutions of distinct types. The order in which these forms have 

 been stated seems to be that of their chronological employment, 

 although the reverse of this order is that of their logical filiation. 



It is proposed in the following paper to exhibit general solu- 

 tions of the class of partial differential equations to which the 

 Laplacian equation belongs, under these three several forms. It 

 is obvious that, as this class of equations contains no term in its 

 right-hand member involving only the independent variables, the 

 general solutions will contain no determinate expression, but will 

 reduce themselves to the arbitrary portions solely. It will be 

 found that the particular solutions employed in the first and 

 second forms are duplicate, each indicating a second in close 

 correlation with itself. In the case of the equation of the sim- 

 plest type, the duplicate solutions are omitted as being evident. 



The instrument employed in arriving at these solutions is 

 known by the name of the Calculus of Imaginaries*, under 

 which arc included the symbols known as Duplets, Triplets, 

 Quaternions, &c., which have of late occupied so considerable a 

 share of the attention of the mathematical world. The laws of 

 the system of Triplets here employed are analogous to those which 

 govern Quaternions. The writer is not aware that any use has 

 as yet been made of these instruments in connexion with the 

 subject of the integration of partial differential equations, and 

 believes the forms of the solutions themselves, as found by them, 

 to be new. It will be seen that he has commenced with the 

 equation of the simplest character, the solution of which is 

 familiar to the reader, and by successively engaging the equa- 

 tions as they rise in order, has endeavoured to show the identity 

 of the method applied to all. 



1 . It is known that a general solution of the equation 



s+s- » 



is given by 



U=SAe w, >* +w '3 y , 



where m { , ?w 2 are connected by the relation 



Now, by the ordinary Calculus of Imaginaries, or Duplets, this 



* Vide " Lectures on Quaternions," by Sir William Rowan Hamilton, 

 recently published by Hodges and Smith, Dublin. 



