Chief Justice Cockle on Quantoids. 347 



two or four of these small films that M. Lamarle obtains the in- 

 complete systems in question. In the case of three of them, 

 resulting from the disappearance of two opposite films, the in- 

 ternal polyhedra which were originally octahedric, are converted 

 into hexahedra of very elegant forms ; the disappearance of two 

 more of the opposed small films converts one of these hexahedra 

 into a tetrahedron of a remarkable appearance, on account of the 

 outlines of its faces. 



XLVIII. On Quantoids. By the Honourable Chief Justice 

 Cockle, President of the Queensland Philosophical Society*. 



L U 



SING a convenient terminology and notation, we may 

 say that the quantoid 



<*<&}> 0|4 % • • a Jifa> l ) n y=yn ... (i) 



has the quadricriticoid 



a?-a,+ ^=n, (2) 



and the cubicriticoid 



2 fll 3-3«^ + « 3 -gj=A; .... (3) 



and that if y n be transformed by the factorial substitution 



y^uY (4) 



into the quantoid 



A (l, A„ A 2 , . . A„ji 1)"Y=Y„ . . (5) 



there subsist the relations 



A =a u, (6) 



and, generally, 



A A w = fl (l, a v a 2 , . . a m J^, A) m u, . . (7) 



or 



A A ;?i = # w m (8) 



2. These criticoids are, as I have shown (Phil. Mag. S. 4. 

 vol. xxiv. pp. 532 & 533), invariable under the factorial substi- 

 tution, and we have 



V-**+'%-t-++%*n .... (9) 



2A 1 3 -3A 1 A 2 + A 3 -^- = 2fl 1 3 -&c. = A. . (10) 



* Communicated by the Author. 



