1190 
Mtl. R. F. GWYTHER ON THE DIFFERENTIAL 
and obtain from it by differentiation the educt of the fifth order 
^yiy-o - ^^yyjyyi- + ^^yi> 
wliich is the difierential invariant of conics, and does not produce any covariant by 
development. 
By this process of eduction, a solution can be found for each order, but it is 
possible to find a simpler series of solutions of = 0 than those thus obtained. 
14. Extei'ision of the theory of eduction of matrices. 
Tf (j!) is a matrix, and if wm write &c. for dfjdx, d^^jdx"^ &c., it is easily seen 
that 
^2^3 = (^^2 — dx + 2) 
Similarly, 
— 2 {2tv — + 1) (f)i. 
no</)3 = 3 (fiv — + 2) ^0, &c., 
n 2 </)„ = n {2iu — d^ + n — 1) 
where lu and d^ are tlie order and degree of cf). 
Hence 
{2w — d^) — {2w — d,,. + 1) 
and 
{2iv — dff — S {2iv — df {2w— d;:^- 2) 2 {2u' — d^-fi 1) (2ir — d^fi- 2) <f)^^ 
are matrices, and are of the form of the matrix and differential invariant of conics. 
Generally, if cf) and i// are twm matrices, and if a and ^ are the respective values of 
2w — dr for each, 
is a matrix. 
Convenient forms of the differential equations and of the irreducible matrices. 
In obtaining the difierential equations of condition (15), and in considering the 
method of eduction by differentiation, there has been a convenience in retaining the 
dijferential coefficients y^, &c., but the forms of the ecjuations 0, and of their 
solutions, are simplified by rejalacing them by a^, a^, &c., where 
= yffn !. 
We thus obtain (15) 
