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LXIII. The Differential Equation of the most general Substi- 

 tution of one Variable. By Captain P. A. MacMahon, R.A.* 



IN the Philosophical Magazine for February 1886, Dr. T. 

 Muir considers the differential equations of the general 

 conic and cubic curves by a perfectly general method. 

 The general linear substitution 



y- 



(a,b){x,l) 



{a',V){x,l) 

 leads, as is well known, to the differential equation 



dx dx 3 \dx 2 J ' 



wherein the expression on the left has been called the 

 Schwarzian derivative : this is a reciprocant ; but it is also an 

 invariant, as may be seen by writing 



a\ 

 dx 



v.t, 



d?y 



^= 2U ' 



#y_ 



dx 



\=V.b, 



when it assumes the form 



12(tb-a 2 ). 



In the case of the general substitution of order n y the 

 resulting expression is no longer a reciprocant, but it is an 

 invariant (catalecticant) of a certain binary quantic f. 



For, writing 



(a,b,c,...)(x,lT _U„ 



y ■ 



we have 



yV n =XJ n . 



Differentiating this equation n + 1, n + 2, n + 3, . . . 2n + 1 times 

 successively by Leibnitz's theorem, and putting 



: y» 



<FY, 



dxP ' JI " dx 

 there results the set of equations :• 



— \r(p) 



— » n ? 



* Communicated by the Author. 



t The formation of the differential equation was recently set as a ques- 

 tion in an examination for Fellowship at Trinity College, Dublin ; but 

 I am not aware that its connexion with the theory of Invariants has been 

 before noticed. 



