182 CHANGE OF THE INDEPENDENT VARIABLE. 



definite by means of the connexion in which it occurs. Thus, 

 for example, as we have stated in Art 170, the brackets 

 expressive of differentiation tinder certain conditions are 

 sometimes omitted, that is, they are left to be suggested by 

 the context. 



In the present case the meaning of the symbols -^- , -^ , 



-J-, ~j- which occur in Arts. 202 and 203 must be carefully 



observed. We might use a more complex notation, as for 

 example the following ; let ty (x, y] be any function of x and 

 y, and let ^ (r, 6} be the form which -\fr (x, y) takes when for 

 x and y we substitute their values in terms of r and 6 ; then 



. , 



dr ( dx } dr { dy ) dr 



and this is the equation which in Art. 202 is expressed more 

 briefly thus, 



du _ du dx du dy 



dr dx dr dy dr ' 



The beginner however must remember that the second 

 form is an abbreviation of the first form, and he should recur 

 to the first form if he has any doubt of the meaning of the 



, , du du du 

 symbols -j- , -3-, -=-. 

 dx dy dr 



It is however with respect to the symbols -r- , j , 



dx dy .. . dr dO 



dt) ' dd which occur in Art. 202, and the symbols -y- , -j- , 



dr d6 



-jjTi -j~, which occur in Art. 203, that mistakes are most 



frequently made. For example, beginners sometimes imagine 



that the ~ of Art. 202 and the -f of Art. 203 are connected 

 dr dx 



ly the formula -j- x -y- = 1. This formula however is quite 



