Page:Mind (New Series) Volume 9.djvu/486

 472 W. R. BOYCE GIBSON : From this we would gather that the property of least action has its intelligible physical import in some obvious proposi- tion as to the conditions under which mechanical work is done. This inference is strengthened by the following extract from the same work : ' Often the phenomena of Nature exhibit maximal or minimal properties, because when these greatest or least properties have been established the causes of all further alteration are removed. The catenary gives the lowest point of the centre of gravity, for the simple reason that when that point has been reached all further descent of the system's parts is impossible. Liquids exclu- sively subjected to the action of molecular forces exhibit a minimum of superficial area, because stable equilibrium can only subsist when the molecular forces are able to effect no further diminution of superficial area. The important thing, therefore, is not the maximum or minimum, but the removal of work ; work being the factor determinative of the alteration. It sounds much less imposing, but is much more elucidatory, much more correct and comprehensive, instead of speaking of the economical tendencies of nature, to say, " So much and so much only occurs as in virtue of the forces and circumstances involved can occur ".' ' Mach again explicitly points out that the physical import of the principle of virtual velocities one of the principles from which the principle of least action can be deduced is simply this same result, that ' motion can never take place except where work can be performed '. Taking this in conjunction with the terms of the extract quoted above it would seem as though this were perhaps the fundamental physical fact of which all mechanical facts are merely differ- ing forms. If so, it would be a most meritorious action on the part of some physicist to point out clearly and without the use of calculus how to deduce from this simple fact that motion never takes place except where work can be performed, that remarkable property of bodies expressed in the law of Least Action. This would completely solve the question of the physical import of the principle.' 2 1 Cf. also Mach, Science of Mechanics, pp. 74-77. 2 It may be that the call for this deduction is a fanciful one. The so- called remarkable property was discovered by Euler as follows : He sought an expression whose variation equated to zero would give the ordinary equations of motion. This expression is, however, as Mach points out, only one of various devisable expressions whose variations equated to zero give the ordinary equations of motion. It does not follow that all these mere mathematical formulae have a direct physical