OF THE HIGHER CALCULUS. 517 



to be symbolised, — the variable operated on, the primary or argument of which it 

 is regarded as a function, and the order of derivation. The relative positions of 

 these symbols is a matter of mere taste or convenience. I have adopted the 

 arrangement of writing the numeral of the order and the primary variable as 

 ante-subponents to the function — thus I use the formula 24 U=Z to denote that Z 

 is the second derivative of U, regarded as a function of t. It is impossible to 

 indicate the proposed relation with fewer letters, and any more would be redun- 

 dant. The same relationship is expressed, according to Leibnitz's notation, by 



d 2 U 



When we desire to indicate the opposite relationship, that is, to state that U 

 is the second primitive of Z, the usual notation is f/Utf — U, whereas by using 

 the ordinary sign of reversion, viz. — , we may indicate the same thing by _ 2t Z = U, 

 that is to say, U is the second primitive of Z, regarded as a function of t. In this 

 way the expressions, etc., 



- 3t x , -2 t x, -i t x, x, lt x, 2tX, 3tX, etc., 



denote a series of functions of t, each one of which is the derivative of the preced- 

 ing, or the primitive of the succeeding term. 



In the general equation nt \J = z, three variable quantities and the constant 

 number n are combined ; and, as I have already said, the most comprehensive 

 form of the problem is " from any three of these to find the fourth." But, since 

 n is necessarily constant, that case of the general problem in which n is the 

 qucesitum, must differ essentially in its nature from the other three ; nay more, 

 unless the given relations among the three variables t, z, v, be such that z is one 

 of the derivatives or primitives of v, regarded as a function of t, the problem can 

 have no solution, it is indeed altogether unmeaning. 



Thus, if nt sin t=f were proposed, that is, if it were demanded, " how many 

 times must sin t be derivated until the result t be arrived at?" our only reply 

 would be, that the question has no meaning, for f is not any derivative of sin t, 

 the only derivatives of this function being cos t, — sin t, — cos t, and + sin t. 



To those who uphold the continuity of algebraic expressions, and claim even 

 for the square root of a negative quantity a real existence, this argument must 

 appear quite inconclusive, because, between the members of the above written 

 series, there may be intermediate terms ; thus, there may be such a term as it as 

 or as nt sc; there may be some yet undiscovered operation, two performances 

 of which may produce the same effect as one derivation ; and although such 

 expressions may appear unmeaning to us, they may not be more so than the 

 analogous symbols #*, a 2i appeared to the earlier algebraists. It is impossible to 

 assert that some such operation may not yet be discovered, although, at present, 

 we can form no conception of its nature. There may even be processes such as 



