426 Proceedings of the Royal Society of Edinburgh. [Sess. 
all vanish, the ternary quadric is equal to 
- (a^ + \x 2 + b 2 x 3 ) 2 ; 
a i 
and if the determinants of 
a Y x-^ 4- a 2 x 2 2 4- a 3 x 3 2 + a 4 a? 4 2 + 2 b 1 x 1 x 2 + 2b 2 x Y x 3 4- 2 b 3 x x x A ) 
+ 2c 1 x 2 x 3 + 2 c 2 x 2 x 4 + 2d 1 x 3 x 4 j , 
aqaq 2 + a 2 x 2 2 + a 3 x 3 2 4- 2 b Y x^x 2 + 2 b^x^ 4- 2cpc 2 x 3 , 
cqaq 2 + a 2 x 2 2 4- a 4 :r 4 2 4- 2 \xpc 2 4- 2b 3 x 1 x i + 2c^x 2 x 4: , 
tqaq 2 4- a 2 x 9 2 + 2bpc Y x 2 , 
apc-f + a 3 x 3 2 4 - 2 > 
apc-p 4" a^x^ 2 4 * 2b 3 xpc^ , 
all vanish, the quaternary quadric is equal to 
— (oqaq + \x 2 + b 2 x 3 4 - b 3 x 4 ) 2 ; 
a i 
and so on generally. An alternative set of conditions is referred to, and is 
exemplified by the case of the ternary quadric, where the vanishing of 
a i G \ ~ i s substituted for the vanishing of 
a Y a 2 a 3 4 - 2b l b 2 c 1 - a — ajb^ - a 3 b 2 , 
this latter being equal to 
{ («i<* 2 - KK a i a s - V) - (<Vi - b AY } a i • 
Sylvester, J. J. (1853). 
[On the conditions necessary and sufficient to be satisfied in order that 
a function of any number of variables may be linearly equivalent 
to a function of any less number of variables. Philos. Magazine, 
v. pp. 119-126: or Collected Math. Papers, i. pp. 587-594.] 
The title at once suggests a connection with Hesse’s converse theorem 
of 1851 (March) : the investigation, however, proceeds on totally different 
lines, and only concerns us because of the doubt thrown on the truth of 
the said theorem by Sylvester’s assertion that the Hessian “ is really foreign 
to the nature ” of the question under discussion. 
Spottiswoode, W. (1853, August). 
[Elementary theorems relating to determinants : second edition, 
rewritten and much enlarged by the author. Crelle’s Journal, 
li. pp. 209-271, 328-381.] 
The latter portion (pp. 343-350) of his chapter (§ ix.) “ On Functional 
