[126] PROFESSOR DE MORGAN ON SOME 



can be satisfied by any values of a, y, y',...y {n \ and is an identical equation. This of itself 

 might be held sufficient to prove that there cannot be even w + 1 constants. 



[Writers* often express themselves as if a given equation of the second order had but one 

 pair of primitives of the first order, having one arbitrary constant in each : and similarly 

 for other orders. But this is not correct : if a = <p (a?, y, y') and 6 = \|/ (x, y, y) be any one 

 pair of primitives, then c =f(cp, \^) is also a primitive of the first order for every form of /. 

 Thus y = ax + b, and a x + by = a 2 + b 2 equally give y" = 0, but the second gives a very 

 different pair of equations of the first order from what the first gives. An equation of the 

 second order represents a family of families of curves. Now though a group is but in one 

 way a group, yet a group of groups is in an infinite number of ways convertible into other 

 groups of groups, differing according to the manner in which individuals from each of the old 

 groups are put together to form one of the new ones. Each of these methods involves a diffe- 

 rential equation of the first order with an arbitrary constant in it. The nature of this last 

 equation dictates the principle on which the groups are formed, the value of the constant 

 settles which group is taken into consideration. 



In the common form y = y'x +fy, it is said that we have only a limited selection from the 

 lines indicated by y" = 0, namely, those contained in y = mx +fm. But if there be an arbi- 

 trary constant in fm, we have the complete solution, since fc is then independent of c in 

 value. The fact is, that a =f(y', y'x — y), which answers to the above with an arbitrary 

 constant in rn, is the general mode of representing an equation of the first order formed from 



y" = o. 



But it is further to be noticed that we are not bound to the entrance of one arbitrary con- 

 stant only in each of the primitives of the first order. The equation /(y' — a, y'x — y — b) = 

 is really satisfied by y" = 0, and by y = Ax + B, provided f(A - a, - B — b) = 0. And this is 

 the complete primitive of / = ; and any other mode 'of integration will introduce the new 

 arbitrary constant only in such combinations with a or b, or both, as will render the three 

 equivalent only to two. 



Many cases have casually occurred in which a greater number of constants enter a 

 primitive than the theory points out ; but it is always found that the constants enter in a 

 manner which reduces their effective number. Nevertheless, it has, I think, been gene- 

 rally received that an integral of the order n — m, of an equation of the wth order, cannot 

 have more than m constants. To shew that this is not always the case, take the equation 

 (A + x) y + Bx + C — 2y = 0. It has three independent constants; and yet it is a primitive 

 of y" = 0. If we differentiate twice, we have {A + x) y'" = 0, and though it cannot be 

 said that three constants disappear in two differentiations, yet it must be admitted that a 

 third constant is driven from the differential equation, which must be y" = 0. Neither do 

 we get four constants in the original primitive; for if we integrate (A + x) y + Bx + C = 2y 

 in the usual way, we find 



y +Ex' + (2AE-B)x+ (a*E- C + BA \ = , 



• The date of this addition is March 17, 1851. 



