[122] PROFESSOR DE MORGAN ON SOME 



f f- y" - *V + #> *j/' - y, y", v — , &c.J = o, 



/ [xY - Y, ~ Y" - XT + Y, ±X\ ~ &c.) = 0, 



depend each on the other ; and one will often be directly integrable when the other is not. 

 This reciprocity of relations always exists when the original assumptions are as in 



V"' _j,<»-i> + ... ± #•-« _ ... and j,<«> =. 



, v I I -si IV y y -ay 

 V wSi ' r " 7r» ' andsoon - 



2.3...W 2.S...»-1 2.3...7&-1 J 2. 3... n 



But there is no case, so far as I can see, which will secure a transformation and chance of 

 integration to every differential equation, 'except the following, which suggested the whole 

 method, and was derived from consideration of M. Chasles's method for equations of three 

 variables. 



Let xy —y=Y, y' = X. Then we have 



* - T, y - XY' - Y, y" Y" = 1, y" Y" 3 + Y'" = o, Y"y" 3 + y'"=o, 



Y" Y v -sY"* v y"y' v -3y'"* 



Y"> ' y" 



Hence the following equations, 



l Y'" Y" Y tv — 3 F'" 2 



f(<»> y-, y'i y"> y'"> y">»-) = °> fiX, xy' - y, x, — , - -^ > - yui >•••) = o> 



depend each on the other. If the second can be solved, then the first is solved by eliminating 

 X between y — XY' - Y, and x = Y' ; and vice versa. If we apply this method to x =/y, 

 y = fy y m x<by + \W> &c. we shall see that it includes most of the common cases in which 

 the solution is reduced to elimination after integration. Thus also y"* + ay" = bxy" 3 is 

 reduced to y" - ay" - by, and y" (xfy + yfy + x2/') - 1 is ultimately linear. 



If we assume x = <p (Y, XY - Y), y = ^ (Y\ XY' - Y) and use a and /3 for Y and 

 XY' - Y as indices of differentiation, we easily obtain 



» = X^ + Xf X + x* . * &c< 



y $«+&■* <p« + <pp-x y" 



This may suggest transformations available in particular cases, and in which the orders of 

 differentiation make their first appearances simultaneously. If we try to carry this further, 

 as by supposing x and y to be each of the form 



<p (Y", XY" - Y, — Y" - XY' + Y) 



we can then express y' by aid of Y", at the highest : but y" requires Y'". 



[Since* the above was communicated, I have found a more general view of the subject, 



* The date of this addition is March 17, 1851. 



