386 Prof. Owen on the Fossil Mammals of Australia. [June 19, 



Part II. Six screws of reference can be chosen, such that each screw 

 is reciprocal to all the rest ; the group is said to consist of coreciprocal 

 screws. The analogy between the convenience obtained by referring the 

 twist coordinates of a rigid body to a group of coreciprocal screws, and 

 the convenience obtained by referring the coordinates of a point to 

 rectangular axes, is pointed out. The important theorem that one screw 

 can be found which is reciprocal to five given screws is discussed. 



Part III. The sexiant is a function of six screws, which can be ex- 

 pressed as a determinant. The propert}?- possessed by six screws when 

 their sexiant vanishes may be enunciated in several different ways ; e. g., 

 wrenches of appropriate magnitudes eqiulibrate when applied about the 

 six screws to a free rigid body. If seven twist velocities about seven 

 screws neutralize, then each twist velocity must be proportional to the 

 sexiant of the six remaining screws. 



Part IV. If a quiescent rigid body receive an impulsive wrench, then 

 the body commences to twist about an instantaneous screw. It is shown 

 that if four impulsive screws lie on a cylindroid, the four instantaneous 

 screws lie on a cylindroid, and also the four impulsive reactions caused 

 by the constraints. The anharmonic ratios of each of these groups of 

 four are all equal. Several special properties of impulsive and instanta- 

 neous screws are also considered. 



Part V. When a body has h degrees of freedom, it is shown that Jc 

 principal screws of kinetic energy can be determined. When an impul- 

 sive wrench is imparted about a principal screw of kinetic energy, the 

 body commences to twist about the same screw. These principles are 

 illustrated by detailed examination of the cases of two and three degrees 

 of freedom. 



Part VI. Miscellaneous propositions. The principal questions dis- 

 cussed are : — the locus plane of a point for twists about the screws on a 

 cylindroid ; the equilibrium of a body under the action of gravity for the 

 different cases of freedom ; remarks on Professor Sylvester's theory of 

 lines in involution ; generalization of a theorem due to M. Chasles. 



VIII. " On the Fossil Mammals of Australia. Family Macro- 

 podid^:. Genera Macropus, Pachysiagon, Leptosiagon, Pro- 

 coptodon, and Palorchestes. — Part IX." By Prof. Owen^F.R.S. 

 Received April 19, 1873. 



(Abstract.) 



In this Part the author concludes his descriptions of the fossils on 

 hand relating to the family of Kangaroos (Macropodidae). He gives ad- 

 ditional evidence of the characters of Macropus Titan, evidence of a larger 

 species of Macropus proper (M. Ferragus). and of two subgeneric modi- 

 fications of that type (Pacliysiagon and Leptosiagon). The characters of 



