850 Mr. J. S. Stuart Glennie on the 
means of Lagrange’s theory, be expressed generally in rational 
terms of (ft fi? fv3 fv4)° ?* 
Denoting those functions by u, v respectively, we have in this 
case vu"), 
A=30, w=, 
Now on substituting w”—! or w° for v in the equation (e,), we 
see that (e,) will merge into 
. Ms Mg Ut Mgt? + 66+ (Mo—])w?=0, . « (e) 
wherein, since (U) is in general irreducible+, we must have 
Hs=0, f4=0,..Ho—1=0. 
Accordingly, on combining the equations (e,), (U), that is to 
say, (e,), (U), we find 
—— 0 . 
u= 953 
the equation (e,) being, as we see, illusory. 
We are therefore not permitted to assume that the resolvent 
product can in general—that is, when (U) has no equal roots— 
be expressed rationally in terms of its fifth power. 
Again, it is generally possible to establish a rational commu- - 
nication between that fifth power and the function W, as is 
evidenced in this latter case from the non-existence of any illusory 
equation corresponding to (e,). 
We are thus furnished, as will be seen, with a new confirmation 
of the validity of my method of solving equations of the fifth 
degree. 
April 1861. 
[To be continued. | 
Mechanics. By J. 8. Stuart Gurnniz, M.A., F.R.AS.T 
Srcrion I. Physics. 
16. | PROCEED to consider the Principles of Energetics, or 
the science of Mechanical Forces, which seem to afford 
the bases of an explanation of physical motions. There is at 
present no attempt at a systematic elaboration of these principles, 
or mathematical application of them to the expression and expla- 
* Tt will be understood that owr present inquiry relates to the possibility 
of expressing either of the two quantities w, » as a rational function of the 
other and of the elements, Aj, Ag,..Am. Thus R(v,..) is supposed to mean 
the same thing as R(v, Aj, Ag,..Am); R indicating a rational function. 
+ As defined by Abel. 
t+ Communicated by the Author. In reference to the first part of this 
paper, note that the word “rotation” in the fourth line from the bottom 
of p. 280 of the last Number for “ revolution.” 
