POINTS OF THE INTEGRAL CALCULUS. [125] 



By a differential equation of the nth order we mean an expression of y M in terms of 

 y in ~ v ,...y, x: and by its complete primitive we mean that relation between y and x which 

 admits of no other relation between y {n) , y ( " -1> , &c. except that in the equation. If we take a 

 primitive which contains n distinct constants (that is, constants which do not admit of reduc- 

 tion to a smaller number by substituting one new letter for a function of two or more), and 

 if we differentiate n times, we know that elimination of the n constants between the primitive 

 and its n differentiated equations will produce one equation, and one only. For the inde- 

 pendence of these n + 1 equations is established by each of them containing a letter which is not 

 to be found in any of the preceding ; the second has y, the third y", and so on. Con- 

 sequently, elimination of n constants by the equations which differentiation furnishes, inde- 

 pendently of all connexion between the meanings of y, y, y", &c, certainly gives only one 

 equation of the nth order. 



But there is a connexion between the meanings of y, y', &c. : taking this into account, is 

 it not possible that some other relation may exist between x, y,...y { "\ not indeed algebraically 

 deducible from the equations, but due to the additional consideration of the connexion in 

 question ? 



Let V be a symbol for the primitive equation, and let V l V 2 ...V n be symbols for the 

 differentiated equations. Let E n be the differential equation deduced from elimination ; and 

 if it be possible, let there be another, F n . Eliminate y^ ni between E n and F„, producing 

 say W n _i. If then we eliminate the n — 1 quantities y\...y (n ~ l) between then equations 

 Pit...P,_ M W„_i> we obtain a relation between x, y, and the n constants, which must be V : 

 for, were it any other, m and y would themselves be constants. Then, since F,...F„_ 1 , W n _ u 

 combined give V , it follows that V Q , V i ,...V n _ 1 combined give W„_ u or then distinct con- 

 stants can be eliminated (algebraically) by help of n distinct equations only; which cannot be. 



Next, observe that if y^ n) be expressible in terms of y {n ~^,...y, x, it follows that y {n + '> is 

 in an infinite number of ways expressible in terms of y {n) ,...y, x. For y {n) = (p (y ( " - ",...y, x) 

 can in an infinite number of ways be made a particular case of y m = \|/ (y^"~ ]) ,...y, x, C), and 

 each of these ways gives its own form of y +l) = ^ (y ln) ,...y, x), true for all values of C. 



From the above, it follows that a differential equation of the nth order cannot have fewer 

 than n constants in its complete primitive, nor more, unless it have an infinite number. By 

 the meaning of the terms, the differential equation is to give the only mode of expressing 

 y {n) in lower orders. But if the complete primitive have n — 1 arbitrary constants only, then 

 y(n-i) can jj e expressed in lower orders, and y (n) can therefore be expressed in lower orders 

 in an infinite number of ways. 



Again, since jf**' can be expressed in lower orders, by the equation itself, it follows that 

 yi,n + h) can b e thus expressed in an infinite number of ways. Hence, there cannot then be 

 n + k constants in the original primitive: for then j/ ( " + * ) could be expressed only in one 

 way. This reasoning applies to all finite values of k, but does not disprove the possibility of 

 an infinite number. 



To settle this remaining point, observe that if the original primitive have an infinite 

 number of constants, it must be possible to assign values to them in such manner as, for a 

 given value of x, to assign given values to y, y' i ...y { " ) . That is to say, the differential equation 



