The Differential Equation of the general Substitution. 543 



Lv.+ (»+2) y „ +1 v«+ i|±|^ * v » )+ - • - + ^Srf ^" )=0 ' 



w.v.+(2»+i)y»vy ) + 2 {yi ) 1 ' )1 y>-.^+- -,.4 g^jj y „ +1 v':> 



or writing 



y l = l!f, */ 2 = 2!a , #3=3! a 1? ... y p =i>! a p _2> 

 these are 



'^ v »" )+a °(^! v » n "' 1)+ai (^2y! v -"" 2)+ - +s »- v - = ' 



a i f Vi- ) +a 1( ^i I? Vi- 1) +« 3 ( -_^ r! V<r 2 > + ...+ a „ V„ =0, 



( B _l)| "" + a3 ( M — 2)! ^*" + • • • + ^o+lVn =0, 



a,iv2°+a, 



Eliminating the w + 1 quantities, 



n\ v »> (»-l)! Vw ' •"' V *' 



between these n -f 1 equations, we find that the desired differ- 

 ential equation is 



t a a x a„_ 2 a n ^ 



a n ^i a n a n +Y 



-2 «2k-1 



= 0. 



This determinant is the catalecticant of the binary quantic 



(t, a , %, . . . a 2 »-i)(X, Y) ; 



and by counting the constants, we see that the general substi- 

 tution is the complete primitive of the differential equation. 



Boyal Military Academy, Woolwich, 

 May 21st, 1887. 



