CHANGE OF THE INDEPENDENT VARIABLE. 181 



Hence, by Art. 202, 



du ~du ..du 



-=cos fl^ + smfl-j- , 

 dr ax ay 



du . ~du n du 



= r sin 6 -7- + r cos a -j- ; 

 dd dx dy 



f du a du 1 . A du i 



therefore -^- cos 6 -, --- sin - rn . 



dx dr r dd i ( . 



du . ,.du \ n du I 

 j- = sm j- + - cos -J2, r , 

 ay rfr r rf^ J 



If we proceed according to Art. 203, \ve must put the 

 equations between x, y, r, 6 t in the form 



dr x x dd y y 



i-| p-p/i _ _ ^ __ _ _ , - / _ ^__ / 



dx <J3? + 2 r' dx~ x z + *~ r 2 



dr _ y _y dd x 



dy~ ^(x a + y*)~r' dy 



,P du _x du y du 



oo; r c/r r 4 c?^ ' 



du _y du x du, 

 ~ 



T* ?/ 



Since - = cos Q and - = sin Q, the formulas (1) and (2) 

 r r 



agree. 



In this branch of the subject beginners are liable to mis- 

 takes from not paying sufficient attention to the precise 

 meaning of the symbols. Generally speaking mathematical 

 notation is so definite that the meaning of any symbol can 

 be settled without regard to the context ; but sometimes in- 

 stead of using a complex symbol to express our meaning 

 without any possibility of mistake we use a symbol which 

 in itself may be ambiguous, but which is rendered perfectly 



