22 Mk DE morgan, ON THE QUESTION, 



examining approved modes of reasoning, that the confusion cannot but be seen to have existed, 

 so soon as the statement of what it consists in is made. 



It is affirmed that the primitive of a primordinal equation cannot have two arbitrary 

 constants : but all that can be proved is that no such differential equation can have two related 

 arbitrary constants in its primitive. 



Let /(a;, y, y') = involve any number of relations between at, y, y : and let (p{x, y, a,b) = 

 be a relation between a and b, or any number of relations. Consequently, selecting one relation 

 by which to satisfy (p = 0, values of a and b can be found to satisfy both (p(x, y, a, h) — 0, and 

 also (p(x + h,y + k, a,b) = 0, for any values of x, y, h, k. Hence, for any values of x and y, 

 y' may have any value whatever: and this is incompatible with /(a;, «/, y') =0. But this is no 

 argument against any form of (p{;v,y, a, b,) = 0, in which the constants are not in relation ; as 

 \^(.r, y, a) . ^(<r, y, b) = 0, For we cannot pretend to satisfy 



\l/{x,y, a) . ^{x, y, b) = 0, %|,(.» + h,y + k, a) .^i^ + h,y + k, b) = 0, 



for any values of w, y, h, k, except by >|/(cr, y, a) = 0, and ^(cT? + h, y + k, b) = 0, or else by 

 \^(.r + h, y + k, a) = 0, ^(x, y, b) = 0. And from neither set can we deduce?/'. If >|/(<if, y, a) = 

 be a primitive oi f{x,y,y') — 0, there appears nothing a priori to prevent our saying that 

 •y|/(.r, y, a) . y]/{x, y, b) = h a primitive. This point will be presently examined. 



It is affirmed that a primordinal differential equation cannot have two really different 

 primitives with an arbitrary constant in each : but all that can be proved is that one prim- 

 ordinal relation cannot have two distinct primitives. If y'=/{x,y) be satisfied by different 

 relations <p[x, y, a) = 0, \|/(a;, y, b) = 0, then, taking a and b so as to satisfy both at a given 

 point (*,«/), we find, generally, two values of y at {x,y). But y'=f{x,y) may give these two 

 values; irreducibly connected, as in y' = 1 ± y/y, or reducibly, as in y' = 1 ± -y/y^ The great 

 point of algebraical interest, namely, that when the two values of y are irreducibly connected 

 (f) = and \|/ = are the alternatives of an equation which can be rationalised or otherwise 

 inverted into % = 0, where ^ is of univocal form, is foreign to the present purpose. That 

 purpose is, to make it clear that the common theorems about the singularity of the constant 

 of integration must be transferred from differential equations to differential relations, of which 

 one equation may contain any number. 



The question whether y = or, which is certainly one relation for determination of y from 

 x, is to be considered as giving one or two relations for determination of x from y, ends in a 

 question of definition, perhaps, but ends in a question which cannot be adequately treated 

 without a close attention to the meaning of the word continuity. And here immediately arises 

 the distinction of permanence of form and continuity of value. 



Form is expression of modus operandi: and permanence of form implies and is implied in 

 permanence of the modus operandi through all values of the quantities to be operated on. 

 In arithmetic, the signs + or — are of the form, and not of the value : but in algebra, the + 

 or — which the ^ei^er carries in its signification are of the value, so called. Accordingly, 

 permanence of form does not necessarily give continuity of value. The immediate passage of 



00 



I smxv.v'^dv from + Itt to — iT, as x passes through 0, might be discovered by the 



•0 



