158 
Transactions of the Royal Society of South Africa. 
four homogeneous integral functions of three variables. It may be 
formulated as follows: If f^, f^, i^, be binary m-thics ; (p^, (p^, 03, ^4 the 
Rosanians of the four possible sets of f's, namely, K(f2, i., f^), — R(fi, fj, f4), 
R(fi, fa, f^), — R(fr, fg, fg) ; (^nd -ijy,, ^/z^, ^ip^ thc Rosanians similarly 
formed from the (p's : then tlie -p's are proportional to tJie f's. The basic 
functions are thus with Rosanes quite special, namely, binary forms all of 
one degree.''' 
The same restriction will be adhered to in the present paper, the 
object of which is to give a fresh and simple presentation of the whole 
subject, following closely on the lines of my corresponding paper on 
Clebsch's theorem in order that the exact extent of the resemblance 
between the two theorems may be made evident. 
3. If Ui, U2, u^, u^ be liomoge^ieoiLS integral functions of the mth degree 
in X, y, and 
i'3 = K(zti, li^, u^), v^=^ — R(?ti, u^, u^); 
then 
u^i\ + u^v^ + u.v.^ + u^v^ = 0. 
This is the same as to say that 
u^ 
^x^y 
w 
'^x'dy 
w 
^x~dy 
'iix'iiy 
^y- 
vhich follows at once from Euler's theorem 
x^.Gol^ + 2xy.Go\^ + y^.Gol^ = m{m — l).coli. 
4. With the same notation 
'd^U^ 
^X^ 
~dx^ 
* In this connection it is worthy of note that the title placed at the top of each page of 
Rosanes' paper by the editor is " Ueber ein die biniiren Formen betreffendes Theorem." 
