of Edinburgh, Session 1875-76. 
93 
4. On the Linear Differential Equation of the Second 
Order. By Professor Tait. 
( A bstract . ) 
This paper contains the substance of investigations made for the 
most part many years ago, but recalled to me during last summer 
by a question started by Sir W. Thomson, connected with Laplace’s 
theory of the tides. 
A comparison is instituted between the results of various pro- 
cesses employed to reduce the general linear differential equation 
•of the second order to a non-linear equation of the first order. 
The relation between these equations seems to be most easily 
shown by the following obvious process, which I lit upon while 
seeking to integrate the reduced equation by finding how the 
arbitrary constant ought to be involved in its integral. 
Let u and v be any functions of x, 
du dv 
dx ^ dx 
A u + Bv 
u + Cv' 
u + Cv 
■ ■ (i), 
where B and A, and therefore their ratio C, are arbitrary constants. 
The elimination of C from (1) must of course give a differential 
equation of the first order in £• 
We have 
_ u" + Cv" A/ + (V \ 2 
* ~~ u + Cv \w + Cv/ 
Now we have, by adding and subtracting multiples of (1), &c., 
j., u + Pu + Qw + C(u -fPu+Qy) /u -\-Cvf ^ 
1 Alf+cL/ - * 
whence, if u and v are independent integrals of the equation 
2/" + iy + Q</ = 0 (2), 
we have the required equation 
£'+£ 2 +PM-Q=o 
and the process above shows why it takes this particular form. 
But (2) gives 
y = A u -f* B# 
