its Analogues, and the Calculus of Imaginaries. 277 



dent of the quantities x, y and z, and their order being equal to 

 the number of the quantities m Li m r In this latter respect it 

 will be found that these solutions are only particular cases of a 

 general law. 



The same remark applies to the solutions of (II.) just found 

 as to the solution of (I.) represented by the formula («j)'. They 

 are not to be regarded as solutions of (II.) in the second form, 

 but only as other shapes of the solutions in the first form, since 

 the limits of the integrals are indeterminate. 



Poisson has furnished an integral of the Laplacian equation 

 strictly in the second form*, namely, 



• 7rf27rf 



1 1 xQ>(y + xsinucosvy' — 1, 

 Jo Jo 



z + x sin u sin py — l)sin u du dv 



4- 



d*C 2v C T , 

 ■g-.-.l I x y v(y + xsmucosvv —I, 



z + x sin u sin v s/ — 1) sin u du dv r 

 but he considers its value lessened by the circumstance of its 

 containing imaginaries under the signs of the arbitrary functions, 

 more especially since, in the application of these solutions, the 

 arbitrary functions ought to be discontinuous. The integral of 

 equation (I.) might have been expressed in a form similar to 

 this, but upon examination it will be seen that it is reducible to 

 the third given form (cj). 



It now remains only 

 equation in the third or 

 given by 



V ~\dz dx J dy) ■\dz+ % dx 1 dy) -U ' 



4ttV=J 



•i (h) 



to find the solution of the Laplacian 

 simple functional form, and this is 



or is 



The 



V = (j>(x + iz, y +jz) + ^{x - iz, y—jz), . , 



the coincidence of which with the first form is obvious, 

 necessity of interpreting all the results of this article, which are 

 analytically complete, is sufficiently apparent, and the practical 

 value of the results mainly depends on their susceptibility of 

 such interpretation. The first form of solution is easily inter- 

 pretable by the ordinary principles of triplets. As regards the 

 two latter, the writer regrets his inability hitherto to satisfy the 

 same demand, and would solicit the attention of those who may 

 favour the present paper with a perusal, to this point. 



It is evident that the solution of the polar form of Laplace's 

 equation is had by substituting in that just exhibited the polar 

 values for x, y and z. 



* Memoires de I'Instiiut, 1818. 



