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consideration of questions of equilibrium, the author calls to his aid 

 a principle, first published by himself in a paper on the ' Theory of 

 the Equilibrium of Bodies in contact,' printed in the fifth volume of 

 the Cambridge Philosophical Transactions, viz. " that when the 

 surfaces of two bodies are in contact under any given pressures, and 

 are in the state bordering upon motion, on those surfaces, then the 

 common direction of the mutual resistances of the surfaces is inchned 

 to their normal at the point of contact at a certain angle, given in 

 terms of the friction of the surfaces by the condition that its tangent 

 is equal to the coefficient cf friction." This angle the author has 

 called "the limiting angle of resistance it has since been used by 

 other writers under the designation of the "slipping angle." 



He next proceeds to determine the modulus of a simple machine, 

 moveable about a cylindrical axis of given dimensions, and acted 

 upon by any number of pressures in the same plane. He applies 

 the principle last stated to determine the general conditions of the 

 equilibrium of these pressures, in the state bordering upon motion 

 by the preponderance of one of them ; and, solving the resulting 

 equation in respect to that one pressure by the aid of Lagrange's 

 theorem, he deduces immediately the modulus from this solution by 

 principles before laid down. The modulus, thus determined, he then 

 verifies by an independent discussion of that particular case in which 

 three pressures only are applied to the machine, one of which has its 

 direction through the centre of the axis. 



This solution he next considers more particularly with reference 

 to a machine moveable about a fixed axis under one moving and one 

 working pressure (their directions being any whatever) and its own 

 weight ; vv^hich last is supposed to act through the centre of the 

 axis. He shows that it is a general condition of the greatest eco- 

 nomy in the working of such a machine, that the moving and work- 

 ing pressures should have their directions, one of them upwards, and 

 the other downwards, and that both should therefore be applied on 

 the same side of the axis of the machine. He moreover shows that 

 if the direction of one of these pressures be given, there is then a 

 certain perpendicular distance of the other from the centre of the 

 axis, and a certain inclination of its direction to the vertical, at 

 which perpendicular distance, and which inclination, this pressure 

 being apphed, the machine will yield a greater amount of work, by 

 the expenditure of a given amount of power, than it will yield under 

 any other circumstances of its application : so that this particular 

 distance and inclination are those whence results the most economi- 

 cal working of the machine. 



Professor Moseley then commences his application of these general 

 principles to elementary machines with the pulley. He establishes 

 the modulus of the pulley under any given inclination of the parts 

 of the cord passing over it, taking into account the friction of the 

 axis, the weight of the pulley and the rigidity of the cord, and adopt- 

 ing, with respect to the last element, the experiments of Coulomb. 

 This general form of the modulus of the pulley he applies, first, to 

 the case in which both strings are parallel, and inclined to the vertical 



