CHANGE OF THE INDEPENDENT VAEIABLE. 183 



inapplicable here ; for it implies that there is a single equa- 

 tion involving x and r and no other variable, which is not 

 the case here. 



In Art. 202 we suppose that x and y are expressed as 



(tJC 



functions of r and 6 ; and -,- means the differential coefficient 



dr 



of x when r varies but does not vary : and as r varies y will 

 also vary, so that on the whole r, x, and y vary, and 6 does 

 not vary. In Art. 203 we suppose that r and 6 are expressed 



dT 



as functions of x and y ; and -j- means the differential co- 

 efficient of r when as varies but y does not vary : and as x 

 varies Q will also vary, so that on the whole x, r, and Q vary, 

 and y does not vary. 



CM or ctii* 



Thus the ^- of Art. 202 and the j- of Art. 203 are formed 

 dr ax 



on different suppositions as to the quantities which vary and 

 the quantities which do not vary. 



Ax 



In the example of the present Article we find that the -=- 



of Art. 202 = cos 0, and the -- of Art. 203 = - = cos 6 ; and 



ax r 



the product of the two is not unity. 



206. Suppose u a function of the three independent vari- 

 ables x, y, 2, and that these are connected by three equations 

 with three new independent variables 6, <f>, r : it is required 



to express -5- . -y- , in terms of differential coefficients 



ax dy dz 



of u taken with respect to the new variables. 



We have, by Art. 174, 



du _ du dO du d<f> du dr^\ 



da; dd dx d(f> dx dr dx I 



du _ du dO du d^ du dr_ ! , . 



dy ~ dd dy dtf> dy dr dy 



du _ du dd du d<j> du dr 



dz ~ dd dz d<f> ~dz dr dz 



