PEOFESSOE CATLET ON TSCHIENHAUSEN'S TEANSFOEMATION. 99 



in the particular case a=e=l, i=(Z=0, c=:Q, that is for the quartic function 

 (1, 0, Q, 0, IJ^a-, 1)^; for, this being so, the equation will be true in general. Writing the 

 equation in the form 



-M0'=U'»(JU'-IH')+4H'«+^% 



and observing that we have 



U'=(B*+D=)+2flBD+49CS 



0'=BD-e, 



4)'=(1-9^^)C(B^-D*), 

 I=l+3fl^ 



and thence 



JU'-IH'=-45='(B^+D*)+(-l-25^-5^)BD+(8^+8^)e, 



the equation becomes 

 -(BD-e)M= 



{-4^(B'+D^)+(-l-2fl^-5fl^)BD+(8QH89*)e} 

 X {B^+D^+2^BD+46C^}^ 

 +4{fl(B^+D^)+(l+fl^)BD-4a^C^}=' 

 +(1-9^)^C^{(B^+D=)^-4B^D''}. 



2. It is found by developing that the right-hand side is in fact divisible by BD — C^ 

 and that the quotient is 



= (_1+10S^_9^XB'+D7 

 +(89+166^-24^')(B^-fD*)BD 

 +(4+8>+4^^-16^'')B^D^ 

 +(_645='-192^^)(B^+D=)C* 

 +(16S=_416^-112^)BDC» 



+ (-128^+128^)0*. 



3. This is found to be 



= _ PU'^+12 JU'H'+4IH'=' 



- 8IJU'0' 



which is consequently the value of — M. We have therefore 



_4>'^= JU'^-1U'^H'+4H'^ 



+(PU'^-12JU'H'-4IH'=')0' 



+ 8IJU'0'* 



+16P0^ 

 which is the required identical equation. 



