270 Mr. R. Carmichacl on Laplace's Equation, 



Now, by the Calculus of Triplets, this relation is equivalent to 



(m 3 + foi, +jm 9 ) (m 3 — im x ->? 2 ) = 0, 

 where t m&j are such operations that 



,-*=-l, /=-l, fm-jL 

 Thus w 3 has either of values + (iro, +jm q ), and the solution 

 assumes the form 



+ 



To obtain the form corresponding to (a } ), we assume 



m l = r cos a, wi 2 = r sin a, 

 and the value of V becomes 



^^ e m l nf+m 2 p+i p . rz _j_ ^ e m x x+m^-i p . rz ^ 



where 



i P 



= i coa u+j sm a~\ 



and a general solution of (II.) in the first form is 



{2 A/"i»+«# { cos V (mf + m 2 2 )s + f« • sin -/(Wj f + w^ 2 )* } 1 

 + h w 



SBc w >* +,B 3J'{cos\/(?w l ? + m 2 2 )2r— ip . sin^(m 1 2 + m 2 2 ).?} J 



the duplicate of which, namely, 



{2Ac^ ro > 8+,,, ^^{cos(m 1 a? + w 2 y) + ip. smfwijay + Wgy)}"] 

 + K w 



2Be^ (fl, »* +w, ^*{cos(wi 1 a? -f w? 2 ?/) — z P . sin^^r + w^) } J 



is had by an obvious modification. 



It may be well to preserve the solutions in these shapes, as 

 it is probable that the quantity i P and those corresponding to it 

 bear some relation to the character of the problems whose law is 

 expressed by the equation of which this is the solution. 



As before, substituting for A and B arbitrary functions of m l 

 and m 2 , we can throw these solutions into the following shapes : 



JY<£>(m v 7w 2 )e m >* +m ^. cos\/(7?z 1 2 -f- m£)z . dm^dm^ ] 



v=\ + k («*)' 



JfiPfa, m^e m *+ m *y . sin */ (w, 2 -f m£)z . dm^dm^ J 

 with its duplicate 



I ff^ m \y m z) <x>s{m l x + m$)e* /(m ? +m * 2)z .dm l dm< i 



y=-\ + k w 



\jyty{m lf m 2 ) sin (m J x + mg/)e* / ' m i* +m & z . dm^dm^ 

 the limits of the integrals in both cases being supposed indepen- 



