24 mh de morgan, on the question, 



3. In a solution we may allow change of form, with a given kind of continuity at the 

 junction. If we mean to stipulate nothing whatever about continuity, we may at any value of 

 X leave one curve, and proceed upon another. If we require non-ordinal continuity, we can 

 only do this where two curves join each other. If we require ordinal continuity or continuity 

 of the same order as the equation, we may propound as a solution of y' = ^\/y any number 

 of parabolas with as much of the singular solution y = as lies between their vertices. If we 

 require every degree of continuity, we have, in the case before us, what is tantamount to 

 requiring permanence of form, in its ordinary sense. 



No prepossession derived from ordinary algebra would be offended by a solution which has 

 a continuity of no higher order than the order of the equation itself: which would allow us, 

 on arriving at the singular solution, or connecting curve, to break off from the curve thitherto 

 employed, to proceed along any arc of the connecting curve, and to abandon this last at any 

 chosen point in favour of the ordinary solution which there touches it. 



In the graphical method by which the possibility of a solution is established, that is, by 

 construction of a polygon from Ay = ;^(.r, y) . A.r, with a very small value of Aw, which may 

 be as small as we please in the reasoning, a solution of y = xi^' V) '® shewn to exist : but it 

 may be one of the kind just alluded to. The draughtsman employed to construct such a 

 solution, when his arc of the ordinary curve comes very near the point of contact with the 

 singular solution, cannot undertake to remain on that ordinary curve, without reference to 

 quantities of the second order. The accidents of paper and pencil are casualties of this order, 

 which might divert his arc of solution from the ordinary curve on to the singular solution, 

 might keep it there for a while, and then throw it off upon another ordinary solution. In fact, 

 the solution established a priori has not of necessity permanence of form, but has only 

 continuity of the order of the equation. And this remark applies to equations of all orders. 

 In the case of y = 2y/y, when once a side of the polygon ends on y = 0, the draughtsman can 

 never leave that line again, without constructing one side by help of Ay = (Aaty. 



It may now be affirmed that (y — ax + b) (y + ax + c) = 0, b and c being perfectly 

 independent constants, is a solution of y'^ — a" = ; nothing in the general theory of the 

 primordinal differential relation in any way withstanding. It remains to examine the assertion 

 that the generality of this solution is not restricted by the supposition b = c. 



To a certain extent this assertion is true: no more curves are obtained or included before 

 the limitation than after it. Beyond this point the assertion is not true. The condition b = c 

 belongs to one mode of grouping a solution of y = a with a solution of y = —a: but there 

 is an infinite number of modes in b = (pc. If ordinal continuity be held sufficient, and if 

 <p{x, y, b) = 0, yp{x, y,c)=0 be independent relations satisfying f(x, y, y) = 0, and if P = be 

 the most complete singular solution, then 



P.(p{x,y,b^) . (p{x,y,b,) x|/(.r,y, cj . x/^Or, y, Cj),., =0 



is the most general solution, where 6j, b2,...Ci, c^,.-. are in any number, and of any values. 

 This however is but equivalent to P . (p{x, y, b) . \^(a7, y, c) = with the usual addition 'for any 

 values whatever of b and c'. 



This point will be best illustrated by reference to the biordinal equation and its theory. 

 A primordinal equation belongs to a group or family of curves which may be called of single 



