52 
of the author in respect to the charge which is thus made 
against the use of differential equations in Dynamical ques- 
tions ; on the contrary, I have always been led to believe 
-hat it is, has been, will be, in the language of differential 
equations that the principles of Dynamics have received 
their greatest development and their most useful applica- 
tions. Still I am greatly impressed with the truth, viz., 
that the experience and opinions of so distinguished an 
authority as is Professor Osborne Reynolds on all questions 
relating to the application of mathematics to the Physical 
Sciences, have great weight, and will always command 
careful attention. 
The problem in question is, however, a simple case of 
Harmonic motion, which is admirably explained and illus- 
trated by:— 1. Thomson and Tait (Natural Philosophy, p. 
36), 2. The late Professor Clifford (Elements of Dynamics, 
p. 20), in sufficiently elementary language to suit the com- 
prehension of any student of practical mechanics. 
I may state here that Newton was the first to invent and 
solve this interesting problem in the pages of the Principict, 
and most writers on the dynamics of variable forces since 
Newton’s time have copied his solution, observing that it is 
exactly the case of a heavy body falling in an open shaft 
through the diameter of the earth. 
As Professor Osborne Reynolds has, therefore, proposed 
his solution of this problem with a view of its being inserted 
in our elementary treatises on Dynamics as the simplest and 
best which has hitherto been given to suit the requirements 
of students in practical mechanics, it may not be deemed 
unnecessary, then, to examine its claims to this distinction, 
and, if possible, to divest it of a few of those things which 
tend to increase its complexity and thereby diminish its 
usefulness. 
No exception, however, can be taken either to the princi- 
ples adopted, or the results derived from them, but, no 
