OF THE HIGHER CALCULUS. 521 



body in a resisting medium is affected by its velocity, so that the differential 

 equation must exhibit the second derivative of the position, in terms of that posi- 

 tion, and of the first derivative. 



And again, when the primary itself is involved, we may call the problem 

 adfected. Thus, if we were to propose for investigation the motion of a body 

 which is urged to a fixed point by an attraction proportional directly to the dis- 

 tance, and inversely to the time, the differential equation would take the form 



— 2t x = | , which may be called adfected. The resolution of this equation leads 



to a series of functions, of which the generic character is contained in the equa- 

 tion x + n lt x + t n x = 0. One of these gives the form assumed by a flexible 

 uniform chain suspended by one end and making minute oscillations. 



This faint outline may serve to give an idea of the scope and classification of 

 problems belonging to the third branch of the calculus. It makes room, as is 

 obvious, for many physical problems which have already been resolved, and 

 which have been regarded as belonging to the calculus of integrals. And its 

 chapters would need to be multiplied, for the purpose of including those numerous 

 cases in which two or more functions of the same primary are combined. 



It may naturally be expected that, when broaching the subject of a new and 

 higher branch of the calculus, I should be prepared with numerous exemplifica- 

 tions of its power and utility. However, I am truly in the position of a natu- 

 ralist, who, having stumbled upon some unlooked-for combination of organs, some 

 duck's bill upon the head of a quadruped, is compelled to invent a new genus, a 

 family, even a tribe, to contain his solitary example, and who is nervously anxious 

 to see the propriety of his extensive generalisations verified by the success of his 

 fellow-labourers in the fields of science. 



