100 PEOFESSOR CATLET ON TSCHIRNHAUSEN'S TEAJSTSFOEMATION, 



Article No. 4. — Calculation of the CuMnvariant. 

 4. We have 



j*=ic.e-ac)'-(i®r 



= (H-iI0'){IU'='-3H'^+(12JU'+2IH')0'+P0'»} 

 -(H-JI07 



whence, substituting for — O'^ its value and reducing, we find 



J*=JU'^+0'.fPU'^+0'^(4IJU')+0"(16P— ^-r). 



Article No. 5. — Final expressions of the two Invariants. 



The value of I* has been already mentioned to be I*=IU'='+0'12 JU'+0'l |P, and 

 it hence appears that the values of the two invariants may be written 



!*=(!, 18J, sr^ru', |0')S 



J*=(J, P, 9IJ, -P+54P5rU', I©')'. 



But we have (see Table No. 72 in my " Seventh Memoir on Quantics " t) 



I(aU+6/3H)=(I, 18J, 3I^a, fif 

 J(aU+6i3H)=(J, P, 9IJ, -P+54P^«, Pf; 



so that, writing a=U', /3=|0', we have 



I*=r(U'U+40'H), 



J*=J(U'U+40'H); 



or the function (1, 0, C, IB, (f5X?/^ 1)* obtained from Tschirkhausen's transformation 

 of the equation U=0 has the same invariants with the function U'U+40'H; or, what 

 is the same thing, the equation (1, 0, C, J3, CX^' 1*)=0 is a mere linear transforma- 

 tion of the equation U'U+40H=O ; which is the above-mentioned theorem. 



t PhUosopMcal Transactions, vol. cli. (1861), pp. 277-292. 



