124 Dr. Hargreave on Differential 



the small letters) 



1/ n y(i-p 2 )* **jp 



«= -{x-py)> v = r * w = 7; — ri 5 



a ^a (1— ^2j7 



and No. 3 produces 



1 2? 2/ 



or 



m=^*, # = &(#■— -J, w = yp~ ( - 1+b \ 



and so on. 



The condition which must subsist between [u and v in order 

 that they may be correlated functions is 



du dv du dv fdudv du dv\ ___ 

 dx dp dp dx ^ \dy dp dp dy) " 



To apply the process, we may begin with any integrable form, 

 however simple ; as, for example, 



at=fp, solved by y= ^f~ l xdx = sjrx. 



The Table above given suggests (amongst others) the following 

 cases and solutions : — 



Equation. Solution. ' 



—-f P - 



6p + a(x-Z)=f( — )• P=t( '-£)• 



w V pJ \ p+ ?yJ Vjp+ J 



pn-i + n-lax^f^^Xp. P n + ™y= + (^i} 



P n + nay=f(^)=\ P . jP* , '- , +« Z 5«*+(^/ 



<l>p+a{xp-y)=f((<j)'p + ax)-').p = f^(<j>'p + ax)- 1 )- 



