1888.] 



Presents. 



433 



A = (r, s, tXq,—p) 2 , 



Aj = (a, b, c, dXq,-p) s , 



A 2 = (e,f, fc, h, i>) 4 » . . • . ; 



and the combinations are such that, when the transformations 



A OT _ 2 = m ! (m— 1) C OT _ 3 



are effected, the resulting forms are the same combinations of the 

 quantities C as the leading coefficients of the fundamental covariants 

 of a binary quantic. 



Some of the properties of the irreducible invariants involving 

 differential coefficients of two dependent variables z and z' are 

 obtained ; and in particular it is shown that there is a single simul- 

 taneous irreducible invariant, pg'—p'q .proper to the rank 1, and that 

 there are four such invariants proper to the rank 2. 



The theory of eduction is next considered. A number of eductive 



operators similar to A h (q ^- — p ~ \ are given ; such an operator, 



V eta oi/J 



applied to an absolute invariant, educes another absolute invariant. 

 Some illustrations are given, and some results, the analogues of 

 reversor operations, are obtained by means of successive educts. 



Finally it is shown that the theory of binary forms can be partly 

 connected with the theory of functional invariants. The equations 

 A 8 = = A 4 are satisfied by A , A 1? A 2 , . . . ., so that these quan- 

 tities may be regarded as a succession of binary quantics in q and 

 — p as variables ; and the same equations are characteristic of the 

 simultaneous concomitants of such quantics. The functional invariants 

 can therefore be expressed in terms of these simultaneous con- 

 comitants. 



Presents, March 15, 1888. 



Transactions. 



Batavia : — Bataviaasch Genootschap van Kunsten en Wetenschap- 

 pen. Tijdschrift voor Indische Taal-, Land- en Volkenkunde. 

 Deel XXXII. Aflev. 2. 8vo. Batavia 1888. The Society. 



Cambridge, Mass. : — Harvard College. Museum of Comparative 

 Zoology. Bulletin. Vol. XIII. No. 6. 8vo. Cambridge 1887. 



The Museum. 



Dublin: — Royal Irish Academy. Proceedings (Polite Literature 

 and Antiquities). Vol. II. No. 8. Ditto (Science). Vol. IV. 

 No. 6. 8vo. Dublin 1888 ; Transactions. Vol. XXIX. Parts 

 1-2. 4to. Dublin 1887; " Cunningham Memoirs." No. 4. 4to. 



