[128] PROFESSOR DE MORGAN ON SOME 



primitive of the last is y = Ax* + Bx + C subject to the condition 2C — 2c =/ (b - B, 

 2A -2a), or it is 



y = Aa? + Bx + c + \f (b - B, 2 A - 2a), 



and of this we must find the most general equation of the first order. The usual equations 

 of the first order will be readily seen to be (A and B being the constants for elimination), 



xy -y = Ax 9 - c -\f(b -y +2 Ax, 2 A - 2a), 



V - B 

 xy -2y m - Bx-2c-f(b - B, = 2a) ; 



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if we throw these into the forms U - A = 0, V - B = 0, then V- B = F (U - A) is the 

 general equation of the first order. 



Taking a simple case of this, as U = A, which gives the first of the above equations 

 again, we ought to find from it y" = ; and if we differentiate both sides, we find 



\x -%f (b-y' +2 Ax, 2 A -2a)} (y" - 2 A) = 0. 



Neglecting the alternative offered by the first factor, which gives a singular solution, we 

 have y" - 2 A = 0, y'"= 0. 



I cannot find any condition worthy of note, except the simple directness of the mode of 

 derivation, which distinguishes the ordinary primitives from those which I have given above. 

 It is not even true that the ordinary primitives are the only ones which contain one arbi- 

 trary constant more than those from which they are immediately derived. Thus a = (xy — y) y' 

 is as much a primitive of y" = as a = xy — y or a = y. And it is to be noticed, in refer- 

 ence to the ordinary theory, that the primitives of the order n - \ are the only ones which 

 are independent of each other. Of those of the order n — 2, £ft (» — l) in number, only 

 n - 1 are independent of each other, the rest being deducible from them ; only n - 2 of the 

 order n — 3, and so on. 



From looking at the small number of cases in which the complete series of ordinary 

 primitives can be found, I surmise the following mode of derivation to be universal. 



A, 



go 



Each letter represents an equation with its order subscript. Of the five equations of the 

 fourth order, F t produces none of the third ; E t , one only, (& ; D 4 , two only, 3 , P 3 ; and 

 so on. This process is repeated on each of the groups, until the complete primitive, g , is 

 attained. Looking at the fact that all in the first group are independent, it may be sus- 

 pected, I submit, that any general mode of thus obtaining the successive primitives will 



