86 Ma DE MORGAN, ON THE QUESTION, WHAT IS THE SOLUTION &c. 



equation a primitive containing any number of separate and independent constants. For 

 Aa + Ai <!>(>■», y) + A2 {^G^)^)}'' + ... = cannot give any relation in which one of these constants 

 disappears in favour of y except ^j + ^„ . y' = 0, in which they all disappear. But this is 

 merely formal ; for Ao + Ai <l>(a?,j/) + ... = is but a transformation of some case of <I>(.r, y) = 

 f{Afi,A^,...) or of (p[ic,y,f{Af„Ai,...)^ =0. All we have done, then, amounts to no more 

 than use of the obvious theorem that a single arbitrary constant is equivalent to an arbitrary 

 function of as many arbitrary constants as we please. Moreover, we may prove that P + y' 

 can only be a factor in the differential of one class of forms. If \F(.v,y)Y give M{P + y), 

 nothing but {\f/F(a!, y)\' can give N(P + y) : and F{ar, y) — const, and \l/F{ai,y) = const, are 

 the same equations. 



But it is otherwise with P + y' , P being a function of x,y, y. This occurs, as previously 

 shewn, in the differentiations of two distinct classes of forms. Thus + y" is a factor in 

 \f{aiy' — y)Y ssiA \n \Fy'\'. The equation 



f{xy' -y) = Ao+A, Fy' + A, {Fy']' + ... 

 is one which contains in every sense, formal and quantitative, as many arbitrary constants as we 

 please; and an alteration in the value of one of them, is an alteration in the character of the 

 relation subsisting between wy — y and y'. Nevertheless, it is impossible to get rid of any one 

 constant in favour of y" in any way except one which results in y" = 0, an equation from 

 which all the constants have disappeared. 



Considerations similar to those which have been applied to primordinal equations might 

 also be applied to equations of any order, 



A. DE MORGAN. 



Univeesity College, London, 

 March 29, 1856. 



