OF THE INDEPENDENT VARIABLE. 191 



u 





i - TX- 



I z=u esimi, and 



that dz - ( l -*>* 



lllclt ~ 7 7- ; - TO . 



av (1 + e cos v) 



fj z ? n* d? 



16. If (a 2 -* 2 )^-|^-z = 0, and ^ + 2 / 8 = a 2 , shew 



d*z 

 that a; 2 -j-s z = 0. 



ajr 



17. Transform 



by assuming a + bx = e l . 



18. If z be a function of the two independent variables x 

 and y, and x and y be connected with two new vari- 

 ables r and 6 by means of two equations, shew how to 



d' 2 z d z z d 2 z 



express -5-2, . . , and -j-^, in terms of the new 



variables. 

 For instance, if x = r cos 6, y = r sin 0, shew that 



z 



- rz = A-B cos'20 + C sin 20, 

 dy 



d^z 



Jt4- = -B sin 2^ +(7 cos 20: 

 dxdy 



n d^z ~. 1 (Tz 1 c?^ 



where ^4 +J3 =- T - 9 , A-B=- -j^ + - -r , 

 dr r ad r dr 



19. If x, y, z, and f, ij, , be co-ordinates of the same point 

 P referred to two different rectangular systems, shew 

 that 



