104 MR ROBERT RAAVSON ON 



In consequence of iv2—v = ohemg a very general primi- 

 tivCj it is inferred that equation (7) is a very general diffe 

 rential equation wliich admits of discriminoidal solution. 



6. Boole, in page 285, has, by a reference to one of the 

 dual equations, unnecessarily limited the values of the 

 constants in the equation 



dy / x^ y dz 



so as to admit of a single solution. 



This equation is satisfied by the single solution 



When a = b, the quantity C is introduced by integration. 



7. The elimination of an arbitrary constant between the 

 primitive and its derived differential equation does not 

 disturb the harmony of the conditional equation Q =0. 



Let (C-RO(C-Kz) &c.=:o . ... (9) 



be a primitive where C is an arbitrary constant and R,, 

 Rj, &c. are given functions of x, y, z. 



Differentiate (9); then 



K dy dy ' ) dx 



+ (!|:(R._C) + §(R,-C)} 



+ (^(E.-C) + ^>(R,-C)}|=o.., (,o) 



This equation evidently satisfies the conditional equation 

 (2). 



Eliminate C from (10) by means of equation (9), and 

 there results 



(R.-RJ^&c. (1^.^^ + ^^^,^") 

 C dy dx dx dz dx j 



{'^-■^^ + 'f-' + '^-^'|&c. = o. („) 

 (. dy dx dx dz dx J 



