192 EXAMPLES. 



20. If xr-- + = 0> and x = ye*, shew that 



*z dz 



, . * fdy\* 



21. Given - + - , and + = I, 



shew that =+__. 



22. Transform -^ sec ^ cosec 6 -^ + yn* tan 2 = 0, into an 



equation in -which a; shall be the independent variable, 

 having given x = log (sec 0). 



23. If y = e~ e and a; = sin 6, shew that 



Result -A 



-r, C?*M tfu d'u . f du du 



24. Express -j-g + 2 7 , + -5-3 m terms of -j- , -y-, 



da; 2 dxdy dy 2 ds dt 



where s = e x +#, and < = e~ a! + e~ 1 '. 



r> ii i^u ^ SM d*u du du 



Result s^-T- 



-T^--f 

 ds dsdt 



25. If x = ae cos ^, and y = ae e sin <f>, shew that 

 d~u 



