APPENDIX I. 



PLUCKKR (J.) Fundamental views regarding mechanics. Phil. 

 Trans. (1866), Vol. 156, pp. 361-380. 



The object of this paper is to " connect, in mechanics, 

 translator/ and rotatory movements with each other by a princi- 

 ple in geometry analogous to that of reciprocity." One of the 

 principal theorems is thus enunciated: " Any number of rota- 

 tory forces acting simultaneously, the co-ordinates of the result- 

 ing rotatory force, if there is such a force, if there is not, 

 the co-ordinates of the resulting rotatory dyname, are obtained 

 by adding the co-ordinates of the given rotatory forces. In the 

 case of equilibrium the six sums obtained are equal to zero." 



MANNHEIM (A.) Sur le dlplacement cTun corps solide. Journal 

 de Mathe"matiques, 2" Series, t. xl. (1866). 



To M. Mannheim belongs the credit of having been the 

 first to study geometrically the kinematics of a constrained rigid 

 body from a perfectly general point of view. This paper con- 

 tains the following theorem : 



When a rigid body has freedom of the second order, any 

 point of the body must be displaced on a certain surface, and at 

 any instant all the normals to these surfaces will intersect two 

 straight lines. 



This is easily seen from the Theory of Screws, because any 

 force reciprocal to the cylindroid expressing this freedom must 

 be a normal to all the surfaces belonging to the points on it. 



SPOTTISWOODE (W.) Note sur Vequilibre des forces dans fespace. 



Comptes Rendus; t. Ixvi., pp. 97-103 (January, 1868). 

 If P , &c., P n _i be n forces in equilibrium, and if (o, i) 

 denote the moment of P , PI, then the author proves* that 



P^o, i) + P f (o, 2) + &c. = o, 

 P(i, o) + + P t (i, 2) + &c. = o, 



P(2 t o) + P l (2, l)+ +. ..-O. 



* We may remark that since the moment of two lines is the virtual co-effi- 

 cient of two screws of zero pitch, these equations are given at once by virtual 

 velocities, if we rotate the body round each of the forces in succession. 



