of the Principle of Least Action, 435 



placed on a beam descends as low as possible ; for if any possible 

 different values of the passive forces would enable that weight to 

 descend, it would certainly descend and produce those values. 

 But there is a restriction implied in the words " physically pos- 

 sible," namely, that the energy expended cannot exceed a certain 

 limit consistently with the physical constitution of the mass of 

 matter ; and the function representing the energy expended is 

 not a maximum in the technical sense of the word. It is just 

 the same in the case of the principle of least resistance : each re- 

 sisting force is indeed the least possible ; but its limiting value 

 depends on the constitution of the whole mass, and it is not, 

 properly speaking, a minimum, though the least possible. In 

 my first article I attempted to state the conditions imposed by 

 the physical constitution of the mass; they were framed with 

 the view of avoiding any assumption as to its continuity. They 

 may perhaps seem arbitrary and insufficient ; but whatever dif- 

 ficulty there may be in stating these conditions correctly, it is 

 clear that (as was pointed out for the conditions stated) they 

 must be introduced in the act of expressing the work in terms 

 of the forces ; and this, which is confirmed by the fact that the 

 function representing the work is susceptible of a minimum value, 

 is the reason why results can in general be obtained by the prin- 

 ciple of least action, but only in peculiar cases by the principle 

 of least resistance. 



The principle employed in these articles has a manifest ana- 

 logy to the dynamical principle of least action, so that I have 

 ventured to use the same name and even to call it an extension 

 of that principle, and they may perhaps be cases of a gene- 

 ral law. But it is not my intention at present to attempt 

 to develope this analogy; the object of this imperfect sketch has 

 simply been to show that the distribution of the resistance of 

 matter to the action of force is governed by a law of Least 

 Action, and that that law is capable of being applied to all, 

 and may advantageously be applied to some of the cases where 

 it is required to find that distribution. Unless Bernoulli's 

 principle mentioned above be an instance, I am not aware that 

 such a method has been hitherto used, and it therefore may 

 perhaps be new. From want of opportunity, however, I am un- 

 acquainted with much that has been done on this subject. I 

 especially regret that I have not seen the new edition of the 

 Lecons of Navier, referred to by M. de St. Venant in a paper 

 translated in the Philosophical Magazine for January 1865. 



April 30, 1865. 



Note. 

 If U ? the work done in a body, be conceived to be expressed in 



