DIFFERENTIAL COEFFICIENTS. 47 



,1 t> dy iJa, . i 

 therefore -^- = - 



(7) v = tan~ 1 -. 

 a 



Put - = , therefore y = ta 

 a 



, f 

 therefore 



dy I dz 

 -/- = 2 -j- 

 dx 1 + z dx 



11 g 



a a 



I ~r 



a 



/ _> T . _i od X 3C 



(8) Let y = tan J r - s 5- . 

 a (a 2 - &e 2 ) 



_ 



Put 7-2 ^-V x = z ; therefore v = tan' 1 z, 

 a (a ox) 



dy dy dz 1 dz 



oTiH -,-,'- r ^ ^--- T" _ _ 



f^a; dz dx 1 + 3 2 cZa; ' 



c& 3 (a 2 - a; 2 ) (a 8 - Sx 2 ) + Ga; (3ica 2 - x 3 ) 

 Now T- = - " . 3 ' ,., v - -' , Art. 31, 

 CP; a(a3x) 





And by reduction we find that 



Therefore '1 = ^T^ 



In fact we have from Trigonometry 



tan* 1 



a (a 2 3x 2 ) a 



7 ) f, 



and therefore the value of ? ought to be -, - 



CiflJ tt 



