182 On Differential EqvMions 



the transformation of which has no analogue in the theory 

 of algebraical equations, inasmuch as a quadratic cannot be 

 deprived of its second and last terms simultaneously. We 

 shall thus see whether there is any instance in which the 

 analogy fails to hold between algebra and the differential 

 calculus. 



13. To effect the proposed transformation of (1) I change 

 the independent variable from ^ to ^ and, at the same time, 

 the dependent variable from y to F, the relation 



2/ = uF - - - - (2) 



subsisting between y and Y. The result of these changes is 

 to transform (1) into 



d?(uY) fdt\^ diuY) (dt\^ cT'x ^ , .^, ^ ,^. 

 df \dxj dt \dx) dP ^ ^ ^ ^ 



or 



c Z^ (u Y) d t d^x d (u Y) fdxV y. 



dt'' dx'dt' ' dt ^ ^^ [dtj ^^ ^^ 



] 4. Developing, we have 



d^ (u Y) _ d^ . 9 l!^ ^ F d'u 

 df -^ dt' '^ dt ' dt '^ df 

 dt d^x d(uY) dt d^x dY du dt d^x ^j. 



dx dt'' dt dx df dt d t dx dt?' 



(dx-)^ , ^^. rd x'] 



Hence, (4) ma}^, after division by it, be replaced by 

 dTY f2 du_dt d'x^ dY 

 df "^ [u dt dx' dt') dt 



f 1 d'u 1 du d t d' X c d x^^ ^ y ^ 



'^[il' dt'~u'~di'dx' IW^ ^\~di] j 



15. Hence the conditions of the transformation are 



(5) 



2 du dtd}x_^ .p. 



udtdxdt'~ 

 and 



1 d'u 1 du d t d' X ( ^^'^Y — o V'7^ 



1j/ ~d¥~~Tc"dl' dx' df'^'^^ [~dt) ~ ' ^^ 



which equations I proceed to solve. 



