DIFFERENTIAL AND INTEGRAL CALCULUS. 89 



is the minimum, corresponding to 6 = 0.. AA is a maxi- 

 mum, and BE' also a maximum, being greater than all 

 adjacent ones. When a 2 = 26 2 , P, J3, P' coincide, and BE' 

 becomes the minimum]. 



CHANGE OF THE VARIABLE IN A DIFFERENTIAL 

 EQUATION. 



Differential equations may often be much simplified, 

 and even reduced to a form in which we .know the solu- 

 tion, by a change either of the independent or dependent 

 variable. It is hardly worth while giving the formulae for 

 such changes, as they should always be effected, not by 

 substituting the particular case in the formulae, but by 

 following the method indicated. 



7 7<J 



If an equation involve a?, y, ~ , ~^ , . . . and we wish 



to change the independent variable to 2, z being a given 

 function of x, (or x of z), we have first 

 dy dy dz dy dx 

 dx dz dx dz ' dz 1 



either of these may be used according as z is given in 

 terms of #, or x of z. 



Whence differentiating both sides, with respect to x, 

 we have 



(fy_<fyd*z dz_<Fy dz_dy<z (dz\ 2 d*y 

 dx* ~ dz dx* + dx dz* dx ~ dz dx* + (dx) dz* ' 

 dx d*y dy d*x dx d*y dy d*x 



dz d? ~ Tz d? dz 7z Tz* ~ 1z ^ 



/"\Y - MM 



dx* 



(dx\* dx (dx\ 3 



\dz) \dz) 



= ^d^z d*ydzd*z (dz\* d^y 



dz di? * dz* dx dx* + \dx) dz 3 ' 



dx d*y dy d z x d*x (dx d*y dy d 2 x 



~~' ~ (dz ~tz* ~ ~dz dz* 



o 



'dx\ 5 

 ,Tz) 

 from which the method is sufficiently apparent. 



