90 Prof. Tait on the Importance of 



change are almost as numerous as are the various kinds of 

 change. The more important of them employ forms of the 

 letter D : — viz. d, d, D, A, 8, and y. 



From our present point of view little need be said of A, 

 which is the equivalent of (D — 1) or of (e d l dx — 1), because 

 the changes which it indicates take place by starts and not 

 continuously. Good examples of problems in which it is 

 required are furnished by the successive rebounds of a 

 ball from a plane on which it falls, or by the motion of a 

 light string, loaded at intervals with pellets. 



Various modes of applying the symbol d are exemplified 

 in the equation 



*HSM3)* + (2h 



In the terms dx, dy, dz, the symbol d stands for changes of 

 value (usually small) of quantities treated as independent. 

 In the term dQ, it stands for the whole consequent change of 

 a quantity which is a function of these independents. By 



the factors ( -p ) &c. we represent the rates of increase of Q ; 



per unit of length, in the directions in which x, y, z are 

 respectively measured. The contrast between the native 

 simplicity of the left-hand, and the elaborate artificiality of 

 the right-hand, member of the equation, shows at once the 

 need for improvement. To express the rate of change per 

 unit of length in any other direction, we have to adopt the 

 cumbrous expedient of introducing three direction-cosines, 

 and the result is given in the form 



7 dQ, dQ^ dQ 

 dx dy dz 



The above equation may be read as pointing out, at any 

 one instant, how a function of position varies from point to 

 point. To express the change, at any one place, from one 

 instant to the next, we write in the usual notation 



*MS> 



But if we have to express the changes, from instant to 

 instant, of some property of a point, which is itself subject to 

 an assigned change of position with time, we have to combine 

 these expressions, and to indicate the relation of position to 

 time. Thus we build up the complicated expression 



ch 

 dt 



*=(f)*+(S£*+(Dt* + ©^ 



