180 On Differential Equations 



10. The transformation thus indicated for the third order 

 is possible for equations of any order. For starting with 

 the equation, deprived of its second term and in which n is 

 greater than 3-, 



we obtain by means of Mr. S. S. Greatheed's general for- 

 mulse for the change of the independent variable (in the 

 Cambridge Mathematical Journal, vol. i. pp. 236 — 8) the 

 following results, true for all values of n : 



tT- a ^^ a ^ A- a ^^ -(b) 



where 



T.. { r d^t^^ rdf^y\^ fd''y\y ) , . 



M meaning a multiple of the included quantities, and a, /3, 7, 

 &c, and X, ^ti, v, &c., being subject to the two relations 



a + P -^ 7 + . . = p - - - (^) 



aX+^lu + <^v+.. = n - - -(e) 



11. We are only concerned with the first three terms of 

 the transformed equation. For the first put p = n and 

 subtract (^) from (e). The result is 



a (\— 1) + 13 i^—1) 4- 7 (v—l) 4- . . , = - (r) 



of which, since a, &c., and X, &c., are to be integers, the 

 only available solutions are 



\ = 1, ^ = 1, ^ = 1, . . . a = 1, y3 = 1, 7 = 1, . . . 



consequently 



^^-^{ Idx) j - - - w 



Next, let p=n — 1, then subtracting as before 



a (X—l) + ^ (/a—l) + 7 (v—l) + . . . = 1 - (^) 



the only available solutions .of which are of the form 

 X = 2, /i = 1, I. - 1, . . a = 1, /3 = 1, 7 = 1, . . . 



so that 



.^(dHfdt^^'-^X .. 



^^-^ = ^\dx^ idx) j - - W 



