310 Mr. U. Moon on Poisson's Solution, ^c. [Dec. 13, 



or consisting of the pair of equations 



or of the single equation 

 or of the single equation 



then in every one of these cases we must have 

 0=11 .F(5)^-S . F(^) . Y{p) -I-T . ¥{^\ 



0=Il.F(^).F(g)+T.F(p).F(2) + {R.F(g),^3 + T.F(p).g}F'(^) 



+ V.FO^).F(^)* 



It is also shovs^n with more or less of generality, and it is capable of 

 being shown generally, that if the given equation admit of a complete in- 

 tegral solution containing two arbitrary functions, it will necessarily have 

 two first integrals, each of which will be of the form 



'F(xyzpq)=::(l){/(xyzpq}. 



It might have been added, that if the general equation of the second 

 order and second degree be written 



and it is satisfied by the equation 



^{xyzpq) = 9 {/{xyzpq) }, 



then F and /must severally satisfy all three of the following equations, viz. 



F(^)-^F(^) = 0, 

 Y{x)^- ¥'{z)p - mF'(p) = 0, 

 F(y) + F(%-.zF(^0 = O, 



where 



. l'-V,t.l'^{^s,+ ^rt)l'~'Prs'l-^^r,, 

 = Vrm'' -f Vrtmn -\- Vtii"— P^m - P^j n +JP, 

 0==(2P,.^-P,.5 . l-^l^rt' l^)m 



•^{2Vty-V,t.l-\--?rt)n 



-(P^^^-P,^ + P,). 



The functions /, F, / must satisfy the same conditions, except in the second of the 

 above cases. 



