OF THE INDEPENDENT VAKIABLE. 179 



The numerator of this fraction is 



(I Y* ff Y \ /* f77* 



in 6 -7 + 2 cos -ft r sin } ( cos 6 ^ r sin 8 } 

 do do / \ do / 



/ ft f fjfj* \ / fj'f* \ 



(cos0--^ 2sm0- T 7 l rcosd}( sin 6 j- a + r cos 6 } 

 \ do do / \ do / 



and the denominator is 



,dr 



(cos ^^3 r sin 6) . 

 \ do / 



lf . 



Hence \ve obtain .. . 



dx* s 



.. 



( ~dr . V s 

 ( cos Q -jj. T sm 6 ) 

 \ do / 



202. Let u be a function of the independent variables 

 x and y, say u =f (x, y] ; and suppose x and y functions 

 of two new independent variables r, 6, so that 



It is required to find the values of -r- and -~ in terms of 



dx dy 



differential coefficients of u taken with respect to the new 

 variables. 



If for x and y we substitute their values in terms of r 

 and 6, we make u an explicit function of r and 6. Now, by 

 Art. 169, 



du _ du dx du dy 



dr dx dr dy dr ' 



du _ du dx du dy 



~do = ~dx~do ~dy d~6' 



From these equations -, and -y- can be found. 

 dx dy 



203. If the equations which connect x, y, r, 6, instead of 

 those in Art. 202, are given in the form 



