12 REPORT 1870. 



ellipsoid of equal energy are the normal axes ; the hody would vibrate about each 

 of tliese axes, as about a fixed axis, and its motion is always compounded of vibra- 

 tions about these axes. 



15. The length of the simple pendulum isochronous with the vibration about 

 each normal axis is proportional to the square of the ratio of the corresponding 

 diameter in the ellipsoid of equal energy to that of the momental ellipsoid. 



16. The body is slightly disturbed from its position of rest by rotation about an 

 axis of displacement through an angle of displacement, and also by receiving a 

 small angular velocitj^ about an initial instantaneous axis ; this displacement and 

 velocity may be uniquely resolved into corresponding displacements and angular 

 velocities about the normal axes, and thus the motion of the body is completely 

 determined. 



VI. A Jiiffid Body rotating about ajixed Point, Gravity being the only Force acting. 



17. A plane drawn in the momental ellipsoid conjugate to the vertical through 

 the point of supension is called the conjugate plane. 



18. For small oscillations to be possible, the instantaneous axis must initially lie 

 in the conjugate plane, and it will continue there throughout the motion. 



19. There are two normal axes which are thus constructed. Draw an ellipsoid 

 whose axes are in the same directions as, and proportional to, the squares of those of 

 the momental ellipsoid, the common conjugate diameters of the sections of these 

 ellipsoids made by the conjugate plane are the normal axes. 



20. The normal axes are not at right angles, except when the centre of gravity 

 lies in one of the principal planes, about the point of suspension ; but a vertical 

 plane dra-s\Ti through one normal axis is always perpendicular to a vertical plane 

 drawn through the other normal axis. 



21. The body would vibrate about either of these normal axes as about a fixed 

 axis, and any small oscillation is compounded of simple vibrations about the normal 

 axes. 



Special attention is directed to the theorem of paragraph 19, which contains the 

 solution of the conical pendulum under its most general form. 



On an Unex^lahied Contradiction in Geometry. By W. K. Cliffoed, M.A. 



Observations on BooWs ' Laws of Thovr/7it.' By the late R. Leslie Ellis. 

 Communicated hy tlie Rev. IIobeet Hakley, F.E.S. 



It appenrs to be assumed in Chapter HI. Section 8, that in deriving one concep- 

 tion from another the mind always moves, so to speak, along the line of predica- 

 mcntation, always passes from the genus to the species. No doubt everything stands 

 in relation to something else, as the species to its genus, and consequently the sym- 

 bolical language proposed is in extent perfectly general, that is, it may be applied 

 to all the objects in the universe. But I venture to doubt whether it can express 

 explicitly all the relations between ideas which really exist, all the threads of con- 

 nexion which lead the mind from one to the other. It seems to me that the mind 

 passes from idea to idea in accordance with various principles of suggestion, and 

 that, in correspondence with the difi'erent classes of such principles of suggestion, we 

 ought to recognize different branches of the general theory of inference. This 

 leads me to a further doubt whether logic and the science of quantity can in any 

 way be put in antithesis to one another. From the notion of an apple we may 

 proceed to that of two apples, and so on in a process of aggregation which is the 

 foundation of the science of discrete quantity. Or again, from the notion of an 

 apple we may proceed to that of a red apple ; and this movement of the mind in 

 lined predicamentali is the foundation of ordinary logic. But it is plain a jJriori 

 that there are other principles of suggestion besides these two, and the following 

 considerations lead me to think that there are other exercises of the reasoning 

 faculty than those included in the two sciences here referred to. In the first place, 

 certain inferences not included in the ordinary processes of conversion and syllo- 

 gism were recognized as exceptional cases by the old logicians. Leibnitz has 



