This paper considers certain simple and practically useful properties of Cartesian tensors in three-dimensional space which are irreducible under the three-dimensional rotation group. Ordinary tensor algebra is emphasized throughout and particular use is made of natural tensors having the least rank consistent with belonging to a particular irreducible representation of the rotation group. An arbitrary tensor of rank n may be reduced by first deriving from the tensor all its linearly independent tensors in natural form, and then by embedding these lower-rank tensors in the tensor space of rank n. An explicit reduction of third-rank tensors is given as well as a convenient specification of fourth-and fifth-rank isotropic tensors. A particular classification of the natural tensors is through a Cartesian parentage scheme, which is developed. Some applications of isotropic tensors are given.

A general formulation is given of a method of reduction of Cartesian tensors, by Cartesian tensor operations, to tensors irreducible under the three-dimensional rotation group. The criterion of irreducibility is that a tensor be representable as a traceless symmetric tensor, its reduced or natural form, invariantly embedded in the space of appropriate order. The general formulation exploits the properties of invariant linear mappings between tensor spaces. Considered abstractly, such mappings bring out the structure of the theory and illuminate the relation to spherical tensor theory. On the other hand, any linear invariant mapping between tensor spaces is equivalent to a combination of operations with the elementary invariant tensors U and E. The general abstract formation therefore has a direct operational representation in terms of the ordinary tensor operations of contraction and permutation of indices. An analogous formulation is given for spinors, and the relations between spinors, Cartesian tensors, and spherical tensors is discussed in the language of the present formalism. Lastly, several examples are given as to how the general formalism may be applied to groups other than the rotation group.

The thermal decomposition of dimethyl disulphide has been stitdied in the gaseous state by a static method. The primary reaction, which follows a reproducible induction period, produces one mole of methyl rnercaptan per mole of disulphide, together with a product of low volatility believed to be a thioformaldehyde polymer:There is also a competing reaction producing a large quantity of hydrogen sulphide. 'The remaining volatile products, hydrocarbons of two or more carbon atoms (believed to be chiefly ethylene), free sulphur, polysulphides, and carbon disulphide are formed either by the latter reaction or by the extensive decomposition of products. The decomposition is catalyzed by hydrogen sulphide, and more strongly by the complete reaction mixture. A mechanism is proposed for the main reaction.The present investigation was undertaken as a contribution to our fragmentary understanding of the mechanisms of reactions i~lvolvi~lg the carbonsulphur bond system in organic sulphur compounds. The C-S-S-C bond system of dimethyl disulphide was of special interest. Little is known about the thermal deconlposition of the alkyl disulphides, although the thermal decompositio~l of aryl disulphides has received some study. The work of Schonberg, Mustafa, and Askar (10) suggests that diphenyl disulphide dissociates a t the S-S bond into two free aryl thial radicals. In an early investigation Otto and Rossing (8) found that on distillation a t atmospheric pressure diamyl disulphide (b.p. 248" C.) gradually decomposes into sulphur or sulphur-rich substances and a tarry residue. Bezzi (3) reported that dioctyl disulphide decomposes a t its boiling point of 190" C. a t 15 mm. Faragher, Morrell, and Comay (7) found that decomposition of vaporized naphtha solutio~ls of various alkyl disulphides a t 496" C. produces the corresponding alkyl mercaptan, hydrogen sulphide, free sulphur, alkyl sulphides, thiophenes, and saturated and unsaturated hydrocarbons.In some preliminary experiments in this laboratory it was found by Patrick (9) that dimethyl and diethyl disulphides decompose above 300" C. yielding co~nplex pressure-time curves. The homogeneous decomposition of dimethyl disulphide was found to produce large amounts of mercaptan and hydrogen sulphide. E X P E R I M E N T A L ReagentsT h e dimethyl disulphide, obtained from Eastman I

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The reduction of products of irreducible Cartesian tensors is formulated generally by means of 3-j tensors. These are special cases of the invariant mappings discussed in Part II [J. A. R. Coope and R. F. Snider, J. Math. Phys. 11, 993 (1970)]. The 3-j formalism is first developed for a general group. Then, the 3-j tensors and spinors for the rotation group are discussed in detail, general formulas in terms of elementary invariant tensors being given. The 6-j and higher n-j symbols coincide with the familiar ones. Interrelations between Cartesian and spherical tensor methods are emphasized throughout.