281 
1909-10.] Dr Muir on the Theory of Orthogonants. 
Now this condition is manifestly satisfied by putting £ r = 6 r , but “ la maniere 
la plus generale de la verifier en exprimant les quantites £ en 0 sera de faire 
tr = 
s=n 
@r + \ ^ K-s 
s—1 
V 
a e s ’ 
les indeterminees X etant assujetees a la condition X,. s 
course implies that 
x r 
S= n ~ r 
— Xgr” This of 
and there have thus been obtained in their general form the two sets of 
equations with which Cayley started in his special case. 
For those who may wish to pursue farther this subject of “ automorphic 
transformation ” we may note that the actual expression of the os’s in terms 
of the fs was given by Cayley in a paper dated 24th May 1854,* and that 
he extended his result to a bipartite quadric function in a paper dated 
10th December 1857.f 
Another problem, which in the early history of orthogonants we have 
seen to be of interest, namely, the simultaneous transformation of two 
quadrics, Cayley also dealt with, the first time in 1849 and the second 
in 1857 + 
Sylvester, J. J. (1853). 
[The algebraical theory of the secular-inequality determinantive 
equation generalised. Philos. Magazine , vi. pp. 214-216 : or 
Collected Math. Papers , i. pp. 634-636.] 
The fundamental theorem here is that if 
Xj = ax + a , X 2 
ax + a bx + /3 
bx + f3 cx + y 5 
ax + a bx + (3 dx + S 
bx + p cx + y ex + e 
dx + S ex + e fx + <f> , . . . 
and the coefficients of the highest powers of x in X 1 , X 2 , X 3 , . . . have all 
the same sign, then the roots of X will be all real and will lie respectively in 
the intervals comprised between -j- oo , the successive descending roots of 
X,_, , and — go . The mode of proof is Cauchy’s (1829). 
* Cayley, A., “ Sur la transformation d 5 une fonction quadratique en elle-meme par des 
substitutions lineaires,” Crelle’s Journ.,1 . pp. 288-299 : or Collected Math. Papers, ii. pp. 192- 
201. See also Brioschi in Annali di Sci. Mat. e Fis ., v. pp. 201-206. 
t Cayley, A., “ A Memoir on the Automorphic Linear Transformation of a Bipartite 
Quadric Function,” Philos. Trans. Roy. Soc. London , cxlviii. pp. 39-46 : or Collected Math. 
Papers, ii. pp. 497-505. 
X Cambridge and Dublin Math. Journ., iv. pp. 47-50 ; and Quart. Journ. of Math., ii. 
pp. 192-195 : or Collected Math. Papers, i. pp. 428-431, and iii. pp. ] 29-131, 
